I had exactly the same problem - and also had a problem with my password not being recognised, even though it was correct. I spent two hours on the phone to apple support (and spoke to three different people), who didn't know how to resolve the problem as they hadn't heard of anyone else with this issue. I also spent 5 hours reinstalling BigSur, which made no difference. I was told I would have to purchase an external hard drive to transfer data to, wipe the machine, then reinstall everything. Feeling very frustrated, and not wishing to fork out money for something that wasn't my fault, I did a bit of research (macforum was good) and this seems to work (I haven't had a frozen cursor for a couple of weeks now)...
- When you open the mac, wait a few seconds before typing in your password.
- Let the mac load the desktop fully before attempting to move the cursor or touch the trackpad - I wait about 5 or 6 seconds. (The cursor is always at the top left of the screen on opening)
I hope this is useful - it's a bit annoying this is an issue (and not just for me, it seems), bearing in the mind it's a brand new machine and expensive...!