I have tried this a couple of times before, yes. Luckily, things seemed to have worked out without much trouble on both occasion.
First time I had attempted playing through a game without using starter Pokémon would be in my second run of FireRed version. Aside from battling Gary, as soon as I'd gotten myself a couple of Poké Ball, I wound up capturing Pidgey and decided to give it a shot and see how gameplay would be without using my starter.
Things wound up to be slightly difficult at beginning, as Pidgey didn't precisely have access to much useful moves, but they eventually began to ease when I started to expand my team by capturing more Pokémon. This was sort of an enjoyable experience, I would say.