Philosophically, I do. However, I play a class-based one.
The issue with skills is that they are a subsystem that only loosely ties into classes, and they kind of break the old-school notion of what a class is. In 2e, a thief has thief skills, and a fighter does not. In 3e, a rogue has a bunch of nice class skills and skill points, and a fighter is less well-endowed, but now it's only a shades of gray distinction between the two. An individual fighter could conceivably be better than a rogue at hiding or talking or lockpicking. And anything that opens up noncombat options for the fighter likely pushes it more and more into the rogue's territory.
That's a system issue, not a conceptual one. For example, a system could make skills available to all characters regardless of class, yet still retain specialized abilities supplementing those skills which are exclusive to class. A rogue and a fighter might both have some skill rank in Stealth, for example, but the rogue would have the ability to use Stealth with partial concealment/cover rather than needing total concealment/cover, or the ability to move without a penalty to Stealth, or the ability to slip between cover without exposing themself, and so on.
To me, that's a good thing; I don't care about niche protection. However, if you're attached to the idea that classes are defined by exclusive capabilities, this is a bad thing. To me, the statement you started this topic with is about making the fighter class less of a fighter class.
Why I like niche protection: it establishes a common language and makes it easier to design modules. Those might sound like small things, but they're a really BIG deal.
I think "making the fighter less of a fighter class" is inaccurate for two reasons: (1) I'm not suggesting sacrificing what the fighter is supposed to do well - fight, and (2) I'm not suggesting intruding on territory of other classes.
I suspect point (2) is contentious especially for players of 3e/Pathfinder where there are copious amounts of classes like Marshal, Knight, and Cavalier that have carved out their own niche...a niche which I would argue should be folded back into the fighter class. Yes, IMO the fighter should kill those classes and take their stuff

However, the fighter should not intrude on the niches of iconic classes like barbarian, ranger, and rogue.
The thing about niche is that it depends on the system's support for non-combat activities. I mean, saying the ranger's niche is, for example, "exploration" is IMO too broad. What are the other players doing during exploration, sitting around twiddling their thumbs? The ability to participate meaningfully in a challenge is not the same as being masterful at the challenge; the cleric and fighter can contribute to exploration without trampling on the ranger's niche. That requires robust exploration rules, however, where different characters can contribute to the exploration in different ways. So my suggestions about the fighter are also made with suggestions for what the system should look like.