Tweaks to prefix-log (#18274)

- Explicitly use `mawk` instead of `awk`, since an extension of the
former is used
- Use `fflush` to reduce interleaving the output of different processes
& streams
- Move the `mawk` command to a shell function, instead of writing it
twice
- Look up the `SUPERVISOR_PROCESS_NAME` environment variable in `mawk`,
instead of reading it in the shell & using complex quoting to pass it to
`mawk`

### Pull Request Checklist

<!-- Please read
https://element-hq.github.io/synapse/latest/development/contributing_guide.html
before submitting your pull request -->

* [x] Pull request is based on the develop branch
* [x] Pull request includes a [changelog
file](https://element-hq.github.io/synapse/latest/development/contributing_guide.html#changelog).
The entry should:
- Be a short description of your change which makes sense to users.
"Fixed a bug that prevented receiving messages from other servers."
instead of "Moved X method from `EventStore` to `EventWorkerStore`.".
  - Use markdown where necessary, mostly for `code blocks`.
  - End with either a period (.) or an exclamation mark (!).
  - Start with a capital letter.
- Feel free to credit yourself, by adding a sentence "Contributed by
@github_username." or "Contributed by [Your Name]." to the end of the
entry.
* [x] [Code
style](https://element-hq.github.io/synapse/latest/code_style.html) is
correct
(run the
[linters](https://element-hq.github.io/synapse/latest/development/contributing_guide.html#run-the-linters))

---------

Co-authored-by: Quentin Gliech <quenting@element.io>

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