I'm not sure you really meant "mulligan" in the sense of a putt that ignores the first putt for scores and takes the second putt for score. The basic point you were making was about "learning" from the first putt.
In that sense, I read the reaction of the others to the suggested "second trial" as "not within the Rule of Golf." Well, of course a "mulligan" that replaces the first putt for score is not allowed in a "stipulated round". But in case there is confusion, a "practice stroke" IS expressly allowed after a player has finished a hole, by Rule 7:
7-2. During Round
A player must not make a practice stroke during play of a hole.
Between the play of two holes a player must not make a practice stroke, except that he may practice putting or chipping on or near:
a. the putting green of the hole last played,
b. any practice putting green, or
c. the teeing ground of the next hole to be played in the round, provided a practice stroke is not made from a hazard and does not unduly delay play (Rule 6-7).
Once the hole has been "played", a golfer can put down a second ball and putt it and in fact do that several times so long as it does not delay play "unduly". A little delay is allowed.
So if you meant "mulligan" in the sense of "practice stroke", perhaps you should show your pals the Rules that allow practicing and learning from missed putts on any hole during the round.
Putting Coach and Theorist