This also happened to me on Big Sur, with the alert You do not have permission to open the application "Brother Scanner".
I was able to solve it as follows:
Remove the existing printer from System Preferences.
Uninstall the existing Brother drivers, using the procedure given here: https://help.brother-usa.com/app/answers/detail/a_id/56738/~/uninstall-the-brother-software-and-printer-driver---macintosh-os-x
After a reboot, I reinstalled the Brother software for the latest macOS version 10.15 (even though my Big Sur version 11.5.1).
There are two parts to this: the "Printer Driver" and the "Scanner Driver (ICA)". I installed both.
After another reboot, I added my brother printer anew using the System Preferences (under "Printers & Scanners").
I was then able to use my Brother MFC as a scanner within macOS Preview on macOS Big Sur, as I did before under Catalina.
I've never used the Brother iPrint&Scan software and am very happy using macOS preview for my scanning needs (other than duplex scanning in the document feeder where we have to turn over the block of pages -- please improve that for us!)
The above worked when upgrading my Intel Mac Mini to Big Sur. It also worked with my M1 MacBook Air, which was previously upgraded from an older laptop using the macOS Migration Assistant.