There is no option similar to the mini.iso for Ubuntu 22.04. The new "live server" installer (subiquity) should meet your requirements except for "Has as little in it as possible".
If you love the mini.iso (you are not alone) and want to hear there is hope for the future then there is a Publishing netboot artifacts spec to create subiquity netboot artifacts, possibly including a "micro.iso". The thread has not had much activity.
If you dig around enough there are some 3rd party projects out there. For instance, this ubuntu-server-netboot project tries to recreate the netboot.tar.gz file that was created along with the mini.iso for minimal installs. I have no idea if it works.
In general, the Ubuntu Server Discourse section has the most information about the mini.iso removal. I would characterize this mostly as end users asking for the return of mini.iso and developers saying it should not have been used to begin with. Some samples.
The mini.iso has in fact never been a “supported” install method for Ubuntu, and is not part of the test matrix for Ubuntu releases. And there will not be an equivalent mini.iso image for use with subiquity.
The mini.iso has also never been a supported install method. It is a side effect of the d-i build process which is not QAed, has known limitations, and is not promoted for download by the Ubuntu website or the Ubuntu cdimage website. It is frequently used to create unsupported, and unsupportable, installations that are missing key Ubuntu packages.