Use retry logic of underlying prefect client properly

[cicdw]

I noticed that our clients no longer use the dedicated send(...) methods that we built to help with flaky API performance. This PR re-engages that logic.

As stated in the httpx docs, calling client.request is

Equivalent to:

request = client.build_request(...)
response = client.send(request, ...)

This also resolves an undetected bug in which PREFECT_CLIENT_RETRY_EXTRA_CODES was not being used.

[aaazzam]

[codspeed-hq]

CodSpeed Performance Report

Merging #16992 will not alter performance

_{Comparing better-worker-error-handling (6d954d5) with main (b634c37)}

Summary

✅ 2 untouched benchmarks