For me, it boils down to verisimilitude. A person becomes skilled at a task by a combination of practice and natural talent. D&D has modeled this by adding your ability score modifier (natural talent) to a number representing your character's practice and training (skill points, skill training etc.).
Yes, but the problem with the skill system is that it isn't a matter of practice, that's my point. It is a matter of killing an arbitrary number of monsters, and if you aren't killing monsters you never advance in that skill. What is more you have to be a particular class to learn that skill, or you can never be as good as someone else no matter how much you use it, or how much your group needs to learn it to be an effective adventuring force.
If you just have ability checks and dispense with the skill system, you don't have these problems. You can take a couple weeks of game time to learn a skill, or you can divide skills among party members and become good at them.
If a skill system worked like Skyrim, where it was classless and you gained a small bonus to that skill every time you did it, then you might have a skill system that doesn't break my suspension of disbelief. But I'm not doing that amount of book-keeping either.
The problems with my system just largely depend on people being reasonable about the characters they play. Frankly, if you don't have people willing to play a character, and act like a character, all the rules in the world aren't going to make things any better and will probably make them worse.
Tools and special circumstances can modify this number as well. They have their place. But give a set of tools to a thief and to me, and he's gonna be better at picking locks. Even a thief who is in general less coordinated than I will be better, because he is more practiced. Take away a cross-country runner's running shoes and give me the best equipment in the world, and he's still going to outrun me.
The cross-country runner is going to outrun you because he is fitter. For the thief example though, I suppose that is true... but I just don't find it worth the book-keeping of a skill system to model. The person who is the most dexterous generally picks the locks, and if someone is stronger they generally breaks down the door instead.
In other words, people play their characters and act how their characters would act. If they don't consider themselves thieves, they don't try to pick locks. Generally as well, they just divide up the skills and retain a reasonable amount of skills for their character. The fewer characters there are, the more skills they have.
In the earliest days of D&D, it was just assumed that characters simply knew the skills to survive in a dungeon. I have heard no reports of things getting out of hand and people assuming any of the corner cases that people are mentioning in this thread.