What is the advantage of this tool over, you know, just using Thunderbird or another MUA to copy your emails to the new mailbox?

Thunderbird stops the whole process on first failure. You won't even know from the log file which email is at fault.

I wonder if it includes workarounds for Google IMAP bugs etc. Some years ago I migrated over 10 years worth of emails from Google to Fastmail and I ran into a lot of issues with Google's buggy IMAP implementation. Things like a random 1 in every 1000 email failing to delete so I would have random emails still remaning. I found out it was a known issue which Google pretty much decided they WONTFIX. I later found Fastmail had its own migration tool that (I presume) has workarounds for various bugs in other providers IMAP implentation making it easier than using e.g. Thunderbird to do it. If I could go back in time and use a tool like this that has a way to work around Google's bugs that would have saved a lot of my time.

Thunderbird only supports moving from one mailbox to another, it doesn't support syncing whole namespaces. It also doesn't support syncing with namespace rewriting or syncing flags, timestamps or access rights. It does not allow using different authorization and authentication and is unsuitable for large syncs as you can't resume anything or have a progress bar. It is also pretty slow, because it does one mail at a time and is just not designed for this.

Note that I used the proper IMAP terms. A mailbox is what the layman probably calls a folder or directory, a namespace is what the laymen probably would call a mailbox or account.

You could use Thunderbird, but it’s slow for large mailboxes and multiple accounts. Being cloud-based, Migrate Wizard moves email faster, tracks progress, retries on errors, and supports incremental syncs - all without running a client locally.