Fingerprint reader not working properly on system login

Hi! I have encountered with a bit of a problem while trying to configure the fingerprint reader on my Thinkpad X250. After configuration, I’m not prompted for my finger print.. Eventually I did get it to work, but I don’t think it’s the proper way. This is how.

I followed the Arch wiki page on fprint step by step, but that didn’t work.
https://wiki.archlinux.org/title/fprint#Configuration
I did a little research on PAM Service Configuration files
https://wiki.archlinux.org/title/PAM#PAM_base-stack
By trial and error, find out that to get it working I had to edit system-login file (/etc/pam.d/system-login) instead of system-local-login. Essentially I just added there the line suggested by the wiki

auth sufficient pam_fprintd.so

I think the problem has to do with PAM stacking, because of the sources for the login application:

login -> system-local-login -> system-login -> system-auth

but I don’t understand really why or how I can solve this other than adding the line to the “higher priority” file (system-login instead of system-local-login). I don’t like this solution because system-login is also used when login via ssh, and possibly in a few other applications, and I really don’t want those to ask me for my fingerprint.

Also, I’m having a known issue where after login with my fingerprint, some password dependent functions are not available (in my case Gnome Keyring).
https://wiki.archlinux.org/title/Fingerprint_GUI#Password
This is mildly and annoying and the workaround which envolves using a USB is even more anoying so I can live with it, but I’d like to know if there’s something else that can be done to overcome that problem. Thank you for your time!