Bump Next.js and drop ESLint (#19898)

This PR bumps Next.js and drops ESLint. 

We only had ESLint installed because it came with `next lint`. We don't
actively use ESLint, it was just used in the playgrounds where we test
things out. Dropping this reduces the amount of dependencies we have to
deal with.

Closes: #19867
Closes: #19852

# Test plan

Tested both the `nextjs` and `v3` playgrounds after bumping the Next.js
dependency and getting rid of ESLint, and they still work as expected.