I see a couple problems, neither of which is so egregious that it can't be addressed. First, what if your background skill just never seems to come up? That sucks when you have 8 skills, so it would suck even harder if you just had 3. This might happen if you had a sailor background, frex.
I very much doubt that a broad enough background would never be applicable. Right off the top of my head, for example, I can think of two reasonably common applications for the sailor. A sailor will be skilled both in rope use and in balance! Ultimately, however, the utility of this skill system is determined by the resourcefulness of the player using the skill.
Second, the class skill part really plays into the hands of DM bias or favoritism. Of course DMs shouldn't be biased or play favorites, but it happens.
The purpose of this system is to give adjudication back to the DM. An element of trust is important for any system that relies on such.
There also seems to be some things missing. Race, for example. You could cover that with background, but being a Dwarf Noble almost seems like I'm getting 2 backgrounds.
Personally, I was thinking that the race would be part of the background, but if you prefer to make a distinction, it could be easily done. Since the bonus for the background check is static, though, functionally, it would not matter. Your Dwarf Noble should not have any richer a background than a Human Noble would.
How does this system handle ordinary skill checks like climbing? Everybody has to climb the rope, after all.
Well, the Background Skill might cover it. If so, in most situations, the DM would probably rule that it does not actually need to be rolled (if it is mundane). If background does not cover it, many classes would (I'd say all the martial ones). The Class Skill would be rolled against a DC.
Assuming that a character no skill, whatsoever, in the climbing of ropes, a straight ability check would suffice (which ability would be up to the DM, but most likely Strength.
It occurs to me that this system
would need to address overlap between the Background Skill and the Class Skill, though. In such cases, the Background modifier could be added into the Class Skill check, but ought not be too large. +2 would be a good number, because, effectively, this makes the background a conditional modifier, which seems appropriate (and in line with racial bonuses to skills in both 3e and 4e).