Here's the thing that I think gets to the bottom of all this....whether you are playing alone or with others, if you are submitting your round for a handicap you have to play by all rules for that to be legitimate. You have to have one set of rules tor everyone to follow, so that when that handicap is later used to adjust scores in a competition that number is contrived the same way for everyone. So in a way any round which affects or uses your handicap is linked to a future competition with other people and all rules apply.
In this case, a "breakfast ball" that is used as a mulligan is absolutely against the rules. A ball that is issued as a practice ball would be a bit more gray but still shouldn't be used, as it could still be seen as testing the wind, similar to not being allowed to test the bunker conditions before a hit.
However, if you are not submitting a handicap you can still be playing "golf" and allow everyone involved to hit a mulligan or breakfast ball or practice shot, whatever you want to call it. You are still playing "golf" just like American League teams still play baseball when they use a DH or Canadian Football League teams still play football when they allow receivers to get a head start, or International basketball teams are still playing basketball when they use a different shaped 3-point line. It all still falls under that one sport, regardless of which variations you are using. To state that someone not following every rule to the "T" is not playing "golf" is just pretentious BS that would be enough to drive many casual golfers away from the sport.
It's a sport for crying out loud, and it should be fun. You can change that up a bit and still have fun. Whether that change be allowing one practice shot on the first hole, or scrambling with a partner, or using "gimmes" within a certain distance. It's still golf and is still fair as long as everyone is following those same rules and has the same opportunities.
Obviously, if you are using a handicap or counting towards one, you should follow every rule to the best of your ability so that your score represents something similar to everyone you might compete with at some point. If you don't have markers or your club doesn't sign scorecards, obviously you can't do that. But you should still follow every other rule.
It's not like "golf" is some sacred thing that has never changed and must be played under one specific set of strict rules. It's just that you need rules to be the same for everyone in a competition, even relating to previous rounds that will effect your final score through a handicap.