I've redone a couple of those table types for friends, and it's a real PITA. Easiest way to get a new playing surface is to make a fitted insert (1/4" MDF) with padding/felt and just set it in place on top of the existing table (inside the rail circumference). You lose rail depth, but trying to do anything else is usually much more than you bargained for.
The construction techniques used on those tables are typically designed to be quick and very inexpensive (aka often shoddy), making disassembly extremely difficult and almost impossible to do without destroying something. Often the outer edge of the rail covering is stapled/glued to the underside of the table, and then wrapped up, over, and back under the rail top and is also stapled/glued to the table top under the rail, if that makes sense (and actually constructed in reverse order). Some rails aren't even really fully wrapped at all, but just have vinyl on two sides with the playing surface glued on top of the vinyl (which is glued/stapled to the table). Difficult to describe without corresponding pics, but once you discover how they're done during disassembly, it's a combination of OMG and WTF. Bottom line: the rail generally doesn't come off of the table unless you destroy it, so easily getting down to just the playing surface felt isn't feasible.
After repairing the second one, I decided it's not worth the effort/cost. I just consider them disposable tables -- cheap, functional for awhile, and when they break or need repair, just throw 'em away. Similar to War Games, the only winning move is not to play buy.