Migration assistant with TM won't proceed further

Hello, I have backed up my MacBook Pro 15" 2015 Monterey with Time Machine. After I did the clean install of MacOS (wiped all data in recovery mode with disk utility, installed a OS) and now I want to copy all of my files, and more importantly all the settings, with migration assistant to the new OS (still the same MacBook).

The Migration assistant finds the TM disk (a USB 3.0 external HDD), so I choose it, click continue. Then there a page with "choose from which back up you want to transfer the information". There's only that very one backup I did before doing the clean install of the OS. So I choose that backup and then I click "continue" and that's were I got stuck, nothing happens, the Continue button goes gray (faded) and that's it, no further page, no option to choose what to transfer etc.

Please, what should I do? How to make it proceed further? Where can be the problem?

p.s., the TM backup is there on the external HDD, with all the apps and everything, it can be open in the running OS, but with migration assistant it's doing nothing.

If the migration assistant won't proceed further, sure I can copy the normal data to the Mac, but is there any way to copy all the settings and all the apps too?

Thank you very much for your help.