... like getting a Mulligan if you touch the ball by mistake with your foot or club?
No, like playing however you want and not worrying about rules until rules matter. When they do matter, like in competition, as long as they're the same for everyone then everyone who voluntarily entered that competition follows the same rules. It doesn't really matter what those rules are. I play lots of golf, and it's golf no matter what rules I apply, but I don't compete with anyone but myself. I have no desire to do so - I get plenty of enjoyment competing with the me of yesterday or of last week or of last year. At the end of the day, when someone asks me how I did, they don't ask me what rules I used. I know whether I played well or poorly, and that's how I answer. Then I go home and forget about it until tomorrow. Almost everyone at the golf course where I play does the same thing. Our rules differ, but we're all playing golf, and some of us are even enjoying some friendly competition with friends, by rules to which everyone agreed, and always different than what the R&A or USGA say.
The big hang-up here seems to be disagreements and hard feelings. Over 30 years of running leagues tells me there will always be disagreements and hard feelings when you combine rules (ANY rules) and people, so any attempt to eliminate those things will be unsuccessful. Since you can't run any kind of competition without rules though, you have to decide what you do want your rules to accomplish. All ball fouls can be more black-and-white and lead to fewer disagreements about what happened or the resolution, but it also makes the game more punitive and less fun for people just beginning on their journey through the sport or those who just want to play and have no desire to improve. If, the first time I went to an actual golf course, I hit the ball into a bush and was told I had to drop the ball behind the bush (by rule the bush is almost always between where you have to drop and the hole) and had to hit over or around the bush time after time, I would probably never go back. No disagreements, no arguments, just a game that, when played by the strictest of rules, is no fun for a beginner.
APA, in acknowledgement that rules are needed so everyone competing can be compared (skill level wise and win/loss wise), has endeavored to come up with a set that doesn't make the game so hard that raw beginners want to quit. Plenty still do, but many find the rules at least palatable to the point where they can enjoy themselves and experience some base level of competition. APA has been more successful at that than any other organization with any set of rules, bringing beginners into the sport (the only way I know to actually grow the sport). Do they claim that other organizations should adopt their rule set? No, because the other organizations may have different goals. Claiming there is only one set of official rules is a farce and is quite pretentious. Every set of rules is official once someone decides to use them. Being the first to adopt the "official" rules is a ridiculous claim, no matter who you are. Your rules were already "official". All you did was change them.
Claiming to be the governing body of anything is also a farce - even the APA has trademarked the phrase "The Governing Body of Amateur Pool". It's just a phrase, one no other organization in the USA can use. It's not like there's some omnipotent entity who bestowed that title upon them - they were just the first to trademark it. Claiming to be some "Governing Body" with "Official Rules" is just that - an empty claim. Governing bodies only govern those who agree to be governed by them, and every set of rules is official as soon as someone agrees to use them.