Just go with a Framework 13. Yes it's 13 inch instead of 12 inch. But you will get the best possible Linux laptop out there. And it's also fully repairable and upgradable. I have the latest Framework 13 AMD 7040, yes AMD. And... do not forget to select the new 2.8k display. Allowing for the first time to just select 200% DPI scaling that just works for Linux! So no scaling issues.! https://frame.work/nl/en/products/laptop-diy-13-gen-amd/configuration/new
Framework laptop 13 is just a bit above 1kg. It's 1.3kg... It is worth weighing this choice (you get the joke? hah, guhmm). If you really need something super compact and very light, maybe a old-school "netbook" will do.
Like the GPD WIN Mini (7 inch, that is super compact, right?).. But really get a Framework laptop hehe.
If you can stretch to a 13 incher (no pun intended), LG gram might be your best bet. Around 965 gram weight, reasonably priced, can run Linux without any problem.
Framework is amazing (I have a 7840u 13in) but they're expensive. 100% worth it to me as an enthusiast and IT professional but possibly not for someone less interested in the tech itself.
But it is not quite a laptop, expensive and very powerful - not sure if that suits you.
Linux wise, most of the stuff works (sleep, power profiles, volume buttons, fingerprint reader, face recognition, pen, touchscreen).
Things that don't work are automatic rotate/accelerometer.
I'm super happy with it, running arch, doing development and using VMs.