The putter head mass matters since two putter heads with dif-
fering mass impacting the same ball on the same green at ex-
actly the same impact velocity will send the ball two different
distances, the more massive / weightier putter having greater
momentum and force at impact. This hardly matters unless the
golfer switches putters often, as a basic familiarity with the heft
of the tool is all that is required for instinctive use.
You are correct in this observation provided that the velocities of the standard and heavy headed putters are the same during the putting stroke.
My experience with standard weight and heavy weight putter heads is somewhat different. I find the standard weight putters well weighted for the short, gravity-assisted putting stroke ... plus the necessary force for the backstroke and stop reversal.
The heavier weighted putter heads(including the added shaft-mounted handle counter-weights) substantially increases the inertia that must be overcome. This tends to create added stress to the hands and arms during the stroke which substantially slows down the normal putting stroke. I have to generate additional force to move the heavy putter otherwise my putts are short. This upsets consistency substantially.
The adaptation to convert from standard weight to heavy weight putters creates a compromise to "feel" and "touch". Those who quickly convert to the heavy putter must have anatomical or psychological reasons for the change. Hand-held heavy putters are problematic in my view, except for belly putters that must be heavier in the head to counter the additional shaft length and heavier grips.
I don't think one should go back and forth between standard and heavy putters; too much neuro-muscular difference in their respective putting stroke mechanics.
I agree with all you say, and admit my "hardly matters" comment was a bit off the mark. You're dead right about the added inertia making a "too heavy" putter a problem in starting and transitioning.
You might find a quibble from some designers about the value of some reasonable heft in the backstroke management. Timothy Winey for one believes that a bit of extra heft that challenges the neuro-muscular system in the backstroke substantially sharpens up touch. There is a limit, though. In comparison to a "light" putter, I have no doubt at all that a "heavier" mass is better for touch consistency, and the limit of "too heavy" causes muscle tension and inconsistency of action. The Winey point may lie between what I call "just heavy enough" and "too heavy". I do find that just a note of aggression in the back-and-thru stroke action helps the body hit the timing of touch with a little sharper precision and consistency, but this is difficult to teach and explain exactly what sort of "aggression" I mean. I think it has something to do with the instinctively intended SIZE of the backstroke and the sharpness and definition of the toss-back ballistic impulse to send the putter head into a motion that will coast to its conclusion far enough back under its own steam. An indifferent impulse in the making of the backstroke doesn't seem as sharp and accurate to size the stroke as this sort of aggression at the initial starting back of the stroke. More to learn!