I also tried looking at the Time Machine backup of the old Mac but the new Mac doesn't have permission to see it for some reason.
As for migration assistant, I don't think it allows you to copy a home directory from your old Mac to a subdirectory of your home directory in your new Mac.
So what should I try?
P.S. It seems that I can't use a Time Machine external disk to copy files back and forth, even manually using "cp -rp".
https://support.apple.com/guide/mac-help/transfer-files-mac-...
If one or both Macs are Apple Silicon, try this: https://support.apple.com/guide/mac-help/transfer-files-a-ma...
If it’s extremely large and this will take too long over the network, use an external hard drive, again just using Finder. No need to mess with Time Machine or any of this other stuff people are talking about. It’s just files, that’s what Finder is for.
rsync is also good.