I was worried about this too, but it hasn't been a problem, for several reasons.
1. The skills are all very broad in scope. There are only 17 of them and the average number of skills per character is 4.5 (assuming a uniform distribution of class and race), so the archetypal party of 5 characters has 22.5 skills, easily covering each skill at least once.
2. Most skills are fully available untrained. Skill Training is a +5 bonus which is the difference between a 10 ability score and a 20 ability score. So the wizard may not be trained in History and the cleric didn't pick up Insight but they still do OK on those checks.
3. Skill Training (feat) lets you pick any skill, even one not on your class list. There aren't many good feats at Heroic tier, so my players have been picking up Skill Training feats to get that extra breadth.
4. It turns out, it's OK for some players to shine in certain challenges and take a back seat in others. Nobody expects Mr. Charisma-Dump-Stat-Fighter to do a whole lot during diplomatic negotiations (maybe aid on some Intimidate checks), but that guy is a hoss when the skill challenge calls for pushing a boulder or tossing the halfling up onto a ledge or other physical feats.
5. For skill challenges, I use
http://www.enworld.org/forum/4e-fan...skill-challenge-system-new-version-1-2-a.html which addresses the problem directly by ensuring that each character gets the same number of skill checks.
So, I don't see any reason to change the skill system at this time (it was easier for me to change the challenge rules). As for your proposed house rule,
1. Giving out a free trained skill is pretty strong. That's worth a feat, and a pretty good feat. It probably won't unbalance much, though, if applied to everyone evenly.
2. Eliminating the class skill list seems roughly balanced. It erodes niche protection but shouldn't break any class balance, since most classes already have their most useful skills on their class list. The only good combo I can think of that this allows is warlock's Shadow Walk + Stealth, but that's not so bad (and it was only ever a feat away).
-- 77IM