You are right; in short Dropbear uses <user-home-directory>/.ssh. but OpenWRT uses /etc/dropbear. Keep reading for the full explanation.
OpenWRT, one of the main users to Dropbear, uses /etc/dropbear for the authorized_keys file since it modified their build upstream to specifically use this folder. This change is do to most OpenWRT being single user (root). You can find a little clearer info here about OpenWRT, as I’m not too familiar myself: OpenWRT Forum: Is OpenWRT single User?
With this in mind, the rest of the Dropbear world uses <user-home-directory>/.ssh. Doing so follows OpenSSH norms and improves permissions as only the user in question should be able to see/use the files.
Now, an issue complicating things is that OpenSSH’s ssh-copy-id was built to use /etc/dropbear, as OpenWRT has a strong influence. If you copy a key with that tool, it will not work here. MichaIng tried changing that, but the openssh-portable project seems to be dragging their feet (‘openssh-portable’: GitHub #250).
So, instead of the conventional tools the key needs to be moved and configured manually.