Each piece builds upon the other, so modifying one thing can have unintended consequences.
Only if you deliberately try to break it, or make a really egregious (and thus usually easily spotable in the theoretical) mistake.
I recall trying to build a swashbuckler class, giving flat AC bonuses (a fraction of the PC's level) as they gained levels. It was to make up for the PC's lighter armor and lack of shield. At the higher levels however, they ended up getting ACs skyrocketing, kind of like a monk but not quite as badly.
And yet, every other d20 game, as well as a houserule in UA, as well as several unofficial swashbuckler type classes have done it, and their ACs didn't skyrocket. Sure, if you do something
badly it doesn't work, but that isn't a condemnation of the idea of houseruling the game.
I've seen numerous attempts at low-item gaming fail (due to magic items being very strong - and needed - for AC and saving throws but being much less important for offense).
I don't even know what you mean with this. Either you're saying that the system itself doesn't work
without houserules, which would completely contradict your point, or you're saying that giving someone a gamebreaking magic item is a bad idea. Which is true, but has nothing whatsoever to do with the notion of houseruling the system, or how difficult or easy it is to do.
And let's talk about monsters... using nothing but core rules, I once drew up an unstoppable dire bear/frost giant barbarian. Actually, it wasn't unstoppable. It was a pretty balanced opponent (eg in terms of AC, hit points, saving throws, attack bonus, damage...), and the PCs killed it after a fierce battle, but its grapple check was so high you literally needed a natural 20 to escape. (The PC barbarian wasn't happy about being grappled like this.)
Again; we're talking about houserules here. How is this story even relevent? And yes, bad GMing causes problems. If you don't spot check your antagonists for problematic areas that need to be fixed before you run, then your game could suffer for it. Again; no idea how this relates to the topic.
Any rules change, especially a big ones (like low item gaming or a new class) impinge on so many factors you can easily lose track of them or make a mistake.
My game is routinely run with so many rules changes that it probably isn't even fair to call it D&D anymore. I've never had a problem. You're describing a situation theoretically, but your examples have nothing whatsoever to do with the problem we're talking about, and have much more to do with poor GMing in teneral.
I think coming up with something like a single feat is easy for 3e though. A single feat isn't likely to affect too many subsystems. (There's always exceptions to that rule though. Persistent Spell, Vow of Poverty, etc...)
So your contention is that all of the houserules in UA, all of the houserules in all of the supplements; both official and otherwise, can't possibly exist, or if they do, they're game breakers? Again; I'm not sure exactly what you're trying to say, because your examples don't support your stated point at all.