I don't see why any of those three terms should bother you, although the last one does often get misused.
"MAMIL" describes the ridiculous sight of (usually visibly overweight) middle age blokes who are generally on an expensive and fancy looking road bike and are trying to kid themselves they are still a fit 25 year old athlete, when in reality they look more like a jelly with legs sticking out the bottom. As a bloke, I probably laugh at them even more than most women do, and the ridicule is fully deserved, IMHO.
"The term "sit up and beg" has been in use forever, and describes the upright riding position of a traditional roadster. I use the term, my mum used to use it, everyone knows exactly what it refers to. Why would anyone have a problem with such a description? It isn't even gender-specific!
"BSO", when used correctly, accurately describes the sort of cheap junk bike which is so badly made as to be unuseable, or doesn't remain functional for long. Cycling snobs unfortunately misuse the term, and use BSO to refer to any cheaper bike than the one they ride, which brings it into disrepute. I own certain bikes that some on here would regard as BSO's, even though all my bikes are fully capable of being used reliably if maintained, but their "crime" is they cost less than a weeks' wages new, rather than a month or more of wages. I don't care if a snob thinks my skip-salvage Apollo MTB is a BSO, I know it isn't because whenever I ride it, it always gets me to my destination in working order.
I'm also interested in the origins of various phrases in common use, but unlike those militant feminist types who go looking for "sexism" and any other sort of "ism" in everything they see, hear, or read, I don't want to start a campaign to try to stop anyone else using whatever term they choose. If a few Snowflake types get offended by any phrases I use, that's their problem to deal with. not mine. I won't be changing any of my use of the English language just because some "professional offendees" think I'm sexist/racist/size-ist/ageist etc. They need to get a life and find something that's actually important enough to worry about, like some third-world religious nut-job country managing to get hold of a viable nuclear missile.