I disagree with the current inclusion of equipment proficiencies (medium armor and thieves tools) in PC backgrounds. Backgrounds should give access to skills, different starting equipment/resources, and different faction recommendation (in addition to unique dialogue choices later).
The major problem with the current implementation is that it destroys game balance. 5e is balanced by restricting the equipment available to each class - and this game ignores that fact to it's detriment. Specifically, giving a level 1 wizard access to medium armor is a huge change to game balance. I've seen it in my games, and I've seen it in the forums: magic users with martial equipment proficiencies are far beyond a typical power curve.
Thieves tool proficiency isn't as big of a problem (because you can always reload for chests anyways) but I think it should still be changed. Thieves tools are part of what makes rogues unique (or half-elves flexible). Giving anyone who chooses the lowlife background access seems like it undercuts the importance of the class choice. Worse, because the lowlife background is balanced around "giving" thieves tool proficiency, a character (like a rogue) that picks lowlife is actually getting less from the background that is most thematically appropriate. I think that's a shame.
Feats are less of an issue here - with one exception. The feat that gives heavy armor proficiency at level 4 should be removed. It's a HUGE boost to casters and no boost to martials - making it unbalanced and thematically incoherent. Other feats, such as the one that gives access to thieves tools or new ones that may give crafting proficiencies (or skills), are largely fine. They have less effect in combat and by level 4 it's just less of a concern. An armor feat that gives one "step" up in proficiency would also be appropriate (so light to medium, for example).
I like backgrounds and I think they should stay, I just don't think equipment proficiencies are appropriate, and they're certainly not balanced.
Typos happen. More so on the phone.