I now 2 times experimented with containers and got them working just to see them stop working at the next hard reboot (i.e. power outage) of the router.
First time was with PiHole and this second time I tried with Adguard home instead.
When it happens, the container will start (status is "running" and I can get shell access to the container) but any services on them will not be reachable from the network. Both the DNS and admin interfaces of Pihole and Adguard stopped working. I am using an external USB stick as recommended. Initially I supected that the random old stick I had found to experiment with PiHole was bad, so when I tried again with Adguard, I swapped to a nearly new Sony USB stick that I have much more faith in. I've had it unplugged to fsck the ext4 filesystem using my laptop with no reported errors on the FS.
I can ping the container from the external network, but no application services is reachable inside the container.
I've tried editing the cmd to add -l and -w to the Adguard Home startup to get some more logs from the container, but nothing gets logged either even with logging enabled.