@jerf , the main aim for sound is to have all the right elements in place, the drivers, the firmware and the controlling commands showing no errors. Basically, these elements seem to be in place in this situation as far as it's come to this point, but sound is still failing.
I may be at a significant disadvantage in that I'm unfamiliar with modern dual booting and the effect that MS can have on the machine which may affect a linux installation. I know for example that an MS installation in dual boot can capture the sound device which will interfere with linux's capacity to use it. In post #15 the result suggests that pipewire was controlling the devices, so I don't know that MS was interfering. In any case, this aspect with dual booting is a variable which I cannot factor in due to unfamiliarity.
The following observations may be of interest.
I came across this comment on a debian forum:
This looks like an old laptop (no insult, mine is just as old). Does it have a detachable battery? If so, do a normal shutdown, then remove the battery for a few minutes. Replace the battery and reboot. The reason for this is that sound cards can sometimes go into sleep mode and not wake up. It is possible that hibernate or a previous Windows shutdown has put the card into sleep mode, and the procedure will clear this.
here:
https://forums.debian.net/viewtopic.php?p=832211#p832211
Another suggestion is for the installation, in debian, of the package: firmware-intel-sound, but the dmesg output from your post #20 did not indicate any missing sound firmware. There's no harm in installing it, but I can't say what the linux mint equivalent is, nor whether it would make any difference. I suspect not.
It may be worth trying to boot another live disk, other than linux mint, to see if that can produce sound. For example one might consider the live disks of debian, fedora, ubuntu, MX linux, minios. There are others. The site: distrowatch.com is a reference point for more.
The kernel driver in use on your machine is snd_hda_intel as shown in post #15. Blacklisting the snd_soc_avs and installing firmware-sof-signed made no difference, so presumably they aren't relevant and one is left with focus on just the single driver.
In post #27 there was a suggestion to see if turning off the power saving function made a difference. That result is yet to come.
From this point on of sound failure, one approach is to look into the parameters available for the sound driver snd_hda_intel and experiment with them. It's tedious time-consuming work and there's no promise of success. Perhaps say if you'd like to go down that rabbit hole.