The fewer miners there are, the easier it is to carry out a 51% attack. And the top miners have much more processing power than the average miner. Just a few of the top ones collaborating could carry it off.

That's definitely a problem, and it's the reason other projects are working on ASIC-resistant proof-of-work. I just don't see how the method I described would make things any worse.

I'm just pointing out that limiting active miners to people with fast internet connections means there will be fewer of them.

This used to be an issue with pools but I don't believe this to be a risk so much anymore.

Why not? Pools are still huge, and at least the "ethical" members of a pool could move their processing power to a different pool. If individuals are bigger than pools now, that means even fewer people have to collude to get 51%.

Are there plans/other networks which distribute blockchain storage and mining somehow?

The Ethereum project has some ideas for making it so each node only has to see a subset of transactions, instead of all of them. It's not an easy problem.

They and a few other projects also hope to pull off a "proof-of-stake" system that mostly does away with mining. Done naively that can be attacked, but there appear to be ways to mitigate the problem. Initially, Ethereum is settling for a proof-of-work algorithm that's more resistant to specialized hardware, so big central miners can't get such an advantage.