I play vanilla, would accessibility still get hurt even if I colonize other planets around it? Do they really depend of core sector and its factions?
Not exactly. In vanilla you can colonize, basically anything and anywhere, and make it profitable by building some structures. Low (i.e. "lower") accessibility only makes your industries to export little bit less commodities and that's it. Building your own colonized piece of space far away of the core worlds can theoretically give even higher accessibility, than in a core, as you can colonize all the objects, making higher colony density than core has = higher proximity. More, you can export your commodities to other your colonies, making profits from inner exports, as far as you can create enough demands.
Theoretically, you can satbomb entire core and live on your own elsewhere.
The only thing you should take in consideration - max import/export values. If you build, for example, refining on a same planet the mining, is two things will happen: mining will supply refining, decreasing import demands, leaving room to other commodities to be potentially imported (which is good). On the other hand, that refining will not create demand and will not increase global market value giving you no additional profit from your marketshare of ore/transplutonics. Visa-versa, you can build those on different planets giving more to global markets of ore and transplutonics, but that is mean your planets had to import more, and if combined import\export demand is higher than max import\potential potential (determined by colony size and accessibility), your colonies will start suffer from this.
The other thing, you should remember is that production/demand values are exponential. You can not supply 8 demand from two 4 production planets, you need at least size 8 production to satisfy size 8 demand.
So the answer is "No, your colonies does not necessary depend from core, but if so they have to rely on themselves, which is easy, but not entirely."