Push results in the background

`pusherClient.Push` should not block testing other databases/configs. See https://github.com/FerretDB/dance/actions/runs/12467709173/job/34797483841 for an example.

We either should make `Push` async (with `Close` waiting until all results are pushed) or change the `main` function logic to push all results in the end. In both cases, that could simplify our pinging logic.