As part of future changes to introduce the `net/http`-only server, we'll
introduce the reliance on Go 1.22-only code.

The way that we currently lint/build/etc the project doesn't make this
easy to do so, so we should instead refactor our build steps to make
this easier.

This is the best option I can see right now, but will be happy to tweak
it if we can find an alternative.

I'd not realised that we were - still - using this separate setup just
in GitHub Actions.

To make this cleaner, as well as fit into our new structure for managing
multi-module `Makefile`s, we can create a new `make` task for CI.

Both to fix issues with Go 1.21's detection of `go env GOVERSION`, as
well as generally making it extra explicit that we don't want to
auto-upgrade.