Please rename plugins/system/backuponupdate to its normal name. Akeeba Backup 9 DOES NOT use FOF. This plugin is unrelated to your issue. Renaming it and forgetting to change its name back will cause update issues when Joomla 4.1 is released next month.
Regarding FOF, you can follow the manual workaround.
Download its ZIP file.
Extract it on your computer.
Rename the folder fof to fof40
Upload the fof40 folder into your site's libraries folder so now you have a libraries/fof40 folder
Whatever was using FOF 4 will continue working
Download and install Akeeba Engage 3.0.1 WITHOUT uninstalling your existing version.
Install the FOF 4.1.0 ZIP file you have already downloaded. Now that it's no longer needed it will uninstall itself. If it doesn't, uninstall it manually. You should not have any other extension using FOF 4 on your site. The only other extension of ours using FOF 3 on Joomla 4 is Akeeba Ticket System and that's only until we release Akeeba Ticket System 5 next week which uses core Joomla MVC. After that there will be no Joomla 4–only extension of ours using FOF anymore. As we announced in August, FOF will be retired.
Nicholas K. Dionysopoulos
Lead Developer and Director
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Please keep in mind my timezone and cultural differences when reading my replies. Thank you!