Start using `StreamMatchEvent` in `startSearch()`

Created by: camdencheek

This converts startSearch to use match events rather than resolver events, then fixes all the type errors caused by it.

Currently in draft state because it depends on commits from both #20610 and #20609, so I can't make a clean PR until at least one of those is merged. I'll update this PR once that happens, but until then, you can get an idea of the direction this is going by looking at the last commit.