Every faction offers something, but nothing nearly as OP as ballistic weave. You can use it on army fatigues and battered fedora to get 220 defense. You can wear heavy combat armor on top of all that to get ~340 defense. At that point, there isn't much that is a threat to you.... particularly if you add a bit of lead shielding and find a few poison resist armor pieces. You'll still want a hazmat suit, rad-X, or power armor for heavy radiation though. As others have said, you don't have to side with the faction...just do a few quests.
Yeah, I suspect it was only ever meant to go on regular outfits that take up most armor slots. Almost all of the clothing options that allow weave are like that. It only works on a couple hats and under-outfits...which seems like a mistake. If it was meant to work on that kind of thing, I'd expect they'd have tagged a lot more of the hats and under-outfits. This seems like dev laziness rather than an intended situation. If it was done that way, the best combinations would hit ~160 armor I think, which puts it more in line regular armor use and makes ballistic weave seem much less required.
If you're on PC, you might as well throw in mods to allow arbitrary armor layering and ballistic weave use, and then use as much armor as you feel fair, since the default state already allows for insane OP armor anyway. In my current playthrough, I'm not using ballistic weave at all on myself... but I've layered heavy combat armor (dark metallic re-textured) with the silver shroud outfit. It looks amazing and has me at about 200 def overall, pretty well short of the default 340 mark I could hit by default. It still feels a little OP, but heavy hitters can still knock a good chunk of my health out. I tend to put weave on companion outfits (best use of weave IMO) and sometimes on outfits for special settlers. Vault-Tech guy gets a 110 weaved clean black suit and a 110 weaved bowler hat.