You did well to set bootmgr to be first in the UEFI boot order screen. The next step will be to regenerate your entire BCD with "BCDBoot". It also will replenish the system partition...
However, I want you first to make a backup BCD file & to check that you can access it in the recovery environment. It isn't good to write files directly to the C:\ root folder. So, first let us make a folder named "BU". Either make it using File Explorer, or enter the following at a Command Prompt (Admin)...
MD C:\BU
That makes a folder named BU into the C:\ root folder. After that, the /Export will work...
BCDEdit /Export "C:\BU\BCD Backup"
After that, check whether you can access it at the Command Prompt of external media. Like I posted above...
(a) Boot a Recovery USB drive or a Win 10 Installation media.
If to the Installation media, choose "Repair this computer" (instead of to install).
(c) At the "Choose an option" screen, click "Troubleshoot > Advanced Options > Command Prompt".
(d) At the Command Prompt, type: DIR C:\BU /a
Also type: BCDEdit
Do you see the files listed? If not, I guess try letter D:, in case they have switched. Sometimes, the drive letters are different in the recovery environment. Which letter gave you the files? Did you get output for BCDEdit, (just to prove it is there)?If/when necessary, to recover the backup BCD, you would enter: BCDEdit /Import "C:\BU\BCD Backup"
(e) Close the Command Prompt, & click "Continue" to return.
I can't say it any better than that. After it is done, I'll tell you the rest. But come back with a signed affidavit that it has been done! And I want the broomstick of the wicked witch of the southeast!
We'll have to put a letter onto your system partition, do a BCDBoot, then remove the letter.