Thanks for tips. Unfortunaly that didn't work.

Firstly, I can not use the link https://static.tp-link.com/2020/202009/20200925/Archer%2520AX10(EU)_V1_200827.zip
directly since I get the warning "Please go to TP-Link Offical Website to get products, software and services for your region."

Anyway, the filename in the link was quite interesting since I'm not able to see that listed in the pages for Finland https://www.tp-link.com/fi/support/download/archer-ax10/#Firmware
After replacing "fi" with "uk" I did find the file. The router even accepted but after upgrade goes to the same Rescue state.

Also, the router doesn't accept any other (older) firmwares from the UK site.

Regarding the rescue alternatives, I'm not quite sure but since this router has the HTTP Rescue page I assume it doesn't support TFTP (and would it even provide any "harder" reset than the web page?). I briefly tried it out but it seemed like the router didn't try to access TFPT using 192.168.0.86 for the server.

 

Finally had some more time to dig in to the issue so I went as far as opened up the cover and connected a serial cable (somewhat according to this https://www.youtube.com/watch?v=IHNrnLnLakE ).
One of the last messages in the boot log was about reset button being set.
Not quite sure if this had been the issue all the time since I had tried reset multiple times.
But maybe it was somehow mechanically stuck against the case.
When I made sure the button is not pressed anymore then the router started up properly and even my old settings were there.