The pain and trouble to the score of making a bogey is MUCH WORSE than the benefit to score of making a birdie. Making a score is easiest with the birdies ONLY needed to reach the score (say, five birdies to make a score of 5 under par) and without ANY bogies. Any bogey erases one birdie or at least always requires making up the bogey with a birdie later; any bogey requires making an extra birdie on a hole that was not initially considered a birdie opportunity but on a "difficult hole to birdie"; and any bogey wastes one of the 18 opportunities to reach the required number of birdies by the hole that was bogied and also wastes another hole where the extra birdie has to be made. All together, given a birdie opportunity, if the golfer misses this, he can make a birdie later on a more difficult-to-birdie hole. But if a golfer bogeys a par hole (not one he planned to birdie), he has to make an extra birdie, he has to make the extra birdie on a hole more difficult to birdie than usual, and he is running out of holes to get the bogey erased from the card. So making a bogey or avoiding a bogey is three times more important than making this or that birdie.
As to the uphill-downhill difference in physics as a supposed advantage downhill and a disadvantage uphill and why it is questionable that Tour players actually benefit from the difference:
Tour players have odd ideas in their heads due to golf culture.
One popular "notion" is "to putt the ball to the apex and then gravity will take over and pull the ball the rest of the way downhill to the hole." This notion is fundamentally stupid and reflects first a belief that gravity does not affect the ball any until it reaches a point along its route called the "apex." And second this is assumed incorrectly to always be a point that is uphill above the hole. And the third goofy-think is that wherever the apex might be, if the ball just sort of arrives there and the force of the golfer's putt is fully spent at this point, gravity is just right to, and there is no friction in the surface that prevents, drawing the ball however far from the apex the rest of the way downhill to the hole.
All three of these goofy-thinks are incorrect in the physics of reality. The first notion is OBVIOUSLY stupid. Gravity always is operative and always curves the roll of the ball unless the surface is perfectly level without any tilt out of gravity level.
The second notion is wrong but not so obviously wrong. But in fact nearly half of all putts never have the ball rolling ANY downhill, and roll uphill at every point. If the "apex" is defined as the point of maximum separation perpendicularly from a straight line between ball and hole (the "baseline"), a putt straight uphill has the apex also on the baseline (no separation at all) and there is no downhill in this putt at all. This is a putt from 6 o'clock if the fall line straight uphill is the line on a clock from 6 to 12, with the hole in the center. What about a 4 o'clock putt? This putt's apex will be to the right of the baseline and its separation will depend upon putt distance, green speed, slope steepness and ball delivery pace. Will a 4 o'clock putt have ANY part of the curve where the ball is rolling downhill? This is POSSIBLE, but only when the break curve has the ball crossing the 9-3 line of the clock before curling into the hole, which is always at the intersection of the 6-12 line and the 9-3 line. Hardly ANY putts starting at the 4 o'clock position ever get above the 9-3 line, so there is NO downhill to the putt, with the possible exception of one or two cases at Augusta National on a Sunday.
The third notion is equally goofy, at an intermediate level of obviousness between the above two notions. The grass from the apex downhill to the hole is never so steep that friction won't slow and stop the ball by overcoming whatever force gravity on the slope provides. If that were the case, the person who picked that pin position would get fired, at least if he were the USGA or PGA Tour setup official. Pin positions should never be located where the green speed and slope steepness combination means the grass friction is inadequate to the gravity-on-tilted-green force rolling downhill. So the friction will eventually stop the ball after some downhill rolling once the golfer's force is spent. The length of this stopping action depends upon steepness of slope and green speed. Usually, pins are well inside this wont-stop situation, so there is plenty of friction and balls normally stop BETWEEN the apex and whatever distance remans downhill to the hole. Knowing facts like that requires years studying PUTTING, not physics formulae in books. Consequently, gravity alone is seldom enough to get the job done. Perhaps if the green Stimps 11 as on the Tour this happens with some frequency, but actually it only happens when the green speed is 11 AND the slope is not mild but somewhat steep, perhaps 3% to 4% or more steepness. And it does NOT usually work when even Stimp-11 greens have 0% or 1% or 2% and perhaps 3% slope.
All together, this sort of stupidity fills a lot of golfer noggins on the PGA Tour and that actively PREVENTS all these golfers from acting on accurate, realistic notions. The main notion missing is that all downhill-rolling balls except balls starting from the 12 o'clock position will be curling in a combination of somewhat parallel to 9-3 and somewhat parallel to 6-12, and this combination will be changing to less 9-3 and more 6-12 as the ball slows down. And pins are never supposed to be located on tilts so steep that friction cannot operate to slow and stop the downhill rolling. This means that if the golfer's "read" is based upon goofy-think, he does not know his line's combination of sideways (parallel to 9-3) and downhill might deliver the ball to a point where the force of the golfer's putt strands the ball too far uphill from the hole, and the friction stops the ball before it reaches far enough downhill in its gravity-powered finish. What he NEEDS to know is that his "read" needs to deliver the ball enough sideways to reach the 6-12 straight-downhill fall line as its force is expended but also at a point not too far above the hole for the friction's opposing any down-tilt gravity. And faster putts where the golfer force does not really get exhausted before the break path reaches sideways to the 6-12 fall line will not end up rolling straight downhill into the cup but may still sink if the pace of the ball and crossing path over cup is acceptable. And reconsidering those putts where the ball actually curves onto the straight-downhll path before the ball reaches the top lip of the cup to enter along the 6-12 line, those only happen when the golfer's pace is nearly the LEAST POSSIBLE. Anything faster than this never curls onto the fall line but reaches the hole and enters the hole not at 12 o'clock on the rim, but some other entry path. And finally, if the golfer' pace is not the MINIMAL PACE that curls the ball at the end onto the very 6-12 line of the hole, the ball will curl at the end onto some other fall line parallel to the hole's fall line, either short of the cup or past the cup (a "flat" surface has parallel "equal-elevation contour lines" -- 9-3 orientation -- and also parallel fall lines -- 6-12 orientation -- perpendicular to all contour lines, only one of which passes thru the center of the cup).
For a golfer to benefit from the actual downhill advantage, he would require a mind a) free of the current widely believed goofy-think notions, and b) informed about the actual state of affairs so he has familiarity with what is real and skill / know-how to benefit. And even then, the benefit is only available on straight-downhill putts starting from 12 o'clock or any other putt where the golfer uses the least pace possible.
So the idea that golfers benefit from the downhill advantage is pretty dubious, and comes from academics running off at the mouth without any familiarity with real putting. Typical dilettante junk.
And then, logically, one should consider and compare how the uphill disadvantage usually REALLY operates in the context of actual putting. The uphill disadvantage is only operative below a certain pace, as above this minimum the supposed disadvantage is overwhelmed by the ball's momentum. In real putting, uphill putts very seldom have less than this threshold minimal pace unless the golfer's skill is flawed. But as a matter of intentional avoiding of the uphill disadvantage, nothing special is required other than the usual robust uphill pace.
So on the one hand the supposed downhill advantage is rare and special and only applies apart from balls at 12 o'clock when the delivery pace is unusually and unwisely slow and the golfer does not have weird ideas of what is causing what; whereas the supposed uphill disadvantage requires nothing intentionally different to avoid.
This is typical dilettante thinking of people familiar with academic physics purporting to describe how the academic physics APPLIES to putting, but in fact these folks don't know enough about the putting reality to apply the physics meaningfully. They ASSUME they understand the putting context and that the academic physics simply allows them to tell golfers what the physics means to them for how to putt.
That's the REAL poblem: academic scientists, whether in physics or optometry or biomechanics or any sort of self-contained collection of generalities, thinking they don't need to know much above what most everyone thinks about putting in order to APPLY the generalities of their science to putting. That is always incorrect, and their notions are simply no better than dilettante notions. Good scientists would first become putting experts before offering applied opinions from physics or optometry or motor sports or biomechanics or psychology etc., but they NEVER recognize this need. It's like trying to build and fly an aeroplane by studying only the general principles of aerodynamics physics but without studying and understanding FIRST birds and kites.
Putting Coach and Theorist