I use a Local Account throughout Windows 8.1 to Windows 11 and have no problem signing in to Microsoft Accounts, more than one of those.
They have put more security hoops to jump through, sure, but it is still possible.
Some things have a skip link e.g. for such things as fingerprint, my laptops have no such hardware.
I use an alternative Email address for verification codes, they demand that in some areas of your Microsoft Account.
Control Panel > Credential Manager > Windows Credentials
You may have to delete some entries in there related to Microsoft Accounts. Seems to be 2 entries at least for each Microsoft Account.
The point of this is to get rid of possible wrong entries and they will subsequently be re-generated.
I never sign in to MS Store unless you have to for certain Apps, no need to do so for updates.