I think Mozilla can get pretty far with one of the smaller open source models. Alternatively, they could even just use the models that will inevitably come bundled with the underlying OS, although their challenge then would be in providing a homogenous experience across platforms.

I don't think Mozilla should get into the game of training their own models. If they did I'd bet it's just because they want to capitalize on the hype and try to get those crazy high AI valuations.

But the rate at which even the smaller models are getting better, I think the only competitive advantage for the big AI players would be left in the hosted frontier models that will be extremely jealously guarded and too big to run on-device anyway. The local, on-device models will likely converge to the same level of capabilities, and would be comparable for any of the browsers.