With `send_and_check_request` we'd be blocking both the main loop and
the caller. `send_request` is only going to be blocking on the main loop
when processing the request.

Release Notes:
- N/A

Based on a flamegraph from `perf`/`hotspot`, we are spending 40% of time
redrawing, another 40% of time downloading stuff (i.e. rust toolchain),
and the rest on text rendering, layout and such. This is with Vulkan
Validation (see https://github.com/zed-industries/zed/pull/8044).

I'm also wondering if it would be better with #7758, but regardless we
should have no problem rendering at 60-120 fps and processing user
input. More follow-ups are expected here.