The documentation on the PYIRONSQLCONNECTIONSTRING env var states

This suggests that you can just set the PYIRONSQLCONNECTIONSTRING and be done with it.

However, setting the PYIRONSQLCONNECTIONSTRING actually has no effect.
Looking at

we see that it is only considered, if we also change the sql type from the default 'SQlite' to 'SQLalchemy'.

While adapting the documentation would be an option, in my opinion the implementation should be changed.
It is absolutely possible to provide sql connection strings for postgres, mysql as well, so if pyiron gives the user the opportunity to provide a connection string, the natural assumption will be that it works in all those cases.