F-Droid need to be a system app to update/install apps on it own. For that Privileged Extension needs to be installed, using TWRP or other Android recovery that can flash updates to the system.
F-Droid
F-Droid is an installable catalogue of FOSS (Free and Open Source Software) applications for the Android platform. The client makes it easy to browse, install, and keep track of updates on your device.
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It isn't still possible, AFAIK
With root privilaged extension it is.
On android 13 however it should be able to do it via session installer (google allowed unattened updates) and even without that there is a shizuku plugin which should allow it to work in theory (however the plugin is buggy and works half the time).
I'm unable to make it work with Shizuku without root
Last I knew fdroid didn't support shizuku. Unless, you talk about 3rd party fdroid clients like droidify and neostore, etc. Also, unless xiaomi borked the adb perms(I'm sure they did like they do for accessibility service apps) shizuku can work with wireless adb.