"adb devices" gives "no permission" when on charge-only
Thanks to the pointer provided by Firelord, I was able to solve the issue. The solution might look weird, but works perfectly:

As root, edit /data/property/persist.sys.usb.config. The only line in that file should read:

mass_storage,adb


We all know there is no usb-mass-storage anymore with Kitkat (except maybe rare exceptions). Well, maybe that's the reason nothing "pops up" anymore, offering to mount some resource :)

To revert from this, either:


open the notification area while connected via USB, find the "connected as …" notification, and tap it
or: go to Settings › Storage, tap the "overflow menu button" (the "3 stapled dots"), and tap the (only) entry which reads like "USB connection"


Then select either MTP or PTP, and there you are.

References:


This is the content of /data/property/persist.sys.usb.config on one of my tablets running Android 4.1 (stock)
This setting is mentioned in this answer, saying if you want to eanble ADB, you can setprop persist.sys.usb.config=mass_storage,adb


Postscriptum

Thanks to the context, one could set up some "toggle" using tasker, executing either setprop persist.sys.usb.config=mass_storage,adb (to "get rid of the popups"), setprop persist.sys.usb.config=ptp,adb (to have it back in PTP mode), or setprop persist.sys.usb.config=mtp,adb to re-enable MTP for data transfer. I don't need any of this, as I use adbfs to access my files ;)