For what it's worth, I like the idea of skills, because they help differentiate between PCs with training in one area or another. On the other hand, I think a lot more could be done to make equipment neat or useful, especially mundane equipment.
But with the bounded accuracy idea, I don't know if straight bonuses is the way to go. I think it works better leveraging the advantage or DC aspects of the system. For instance, climbing a wall might be a hard DC, but climbing a wall with a grappling hook and rope is easy. Picking a lock without lockpicks is an epic feat, with lockpicks hard, and with the right key it is so trivially easy that everyone with an intelligence above 3 & opposable thumbs doesn't even have to roll. Advantage and disadvantage could work well to make certain tools special or better than others, for instance, a Dwarven hammer or elven clothing might grant advantage on their respective skill checks.
I just think making items about bonuses leads to more of a Christmas tree effect. Having them adjust DCs makes it about having the right tool for the right job. Making it about advantage or disadvantage makes it worthwhile to hold onto that terrifically useful dwarven hammer or whatever other cool item you found.
Also, the locked door example was totally not inspired by my puppy staring at a door in an effort to figure out how to open it at this very moment.