The best standalone character is falco, he doesn't have too much problems with anyone, just some characters force him to play smarter. In theory, I would say fox would be the best as a standalone, but that has proven impractical. I also think fox has a problem that he can approach any given match so many ways, it is hard to know what play style is best where, let alone be able to use all of his play styles well.
Shiek is probably the best at making lower tier characters irrelevant, but she has a few problems with more relevant characters like puff or ice climbers.
Peach isn't a worse character because her best player retired. You can check the frame data, everything works the same.