Full Disk Access items constantly disappear

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Do not experience this issue on Ventura 13.5.1 on a M1 Mac Mini

Though I do not any Third Party Security Software Installed either

There are no known Windows-like Viruses in the wild that self replicate and affect macOS, because of the underling UNIX  Foundation and Permission Limitation. 

Starting in macOS 11 Big Sur and above. 

The Operating System resides in a Sealed and Read Only Volume that can not be opened by the User nor by Third Party Applications.

The only Entity that can open and modify or alter this Volume is Apple.

That would occur when a update or UpGrade is performed.

The Only thing this Antivirus software is protecting is the Bank Account of the Developers and for zero return to the User aside from the problems this software creates.

AntiVirus Developers purposefully Market their product to create a “ Fear Factor “. In so doing, creating a False Need for their Product.

The The Built in Security  is all that is required to protect the computer.

Any of the below should be removed as per Developers Instructions 

1 - Disk Cleaner 

This will include CleanMyMac , aka “ BrickMyMac “

MacKeeper,

Dr Cleaner Pro Plus

Open Cleaner one Pro

Macmechanic

Clean my Mac x.

2  - Third Party Security Software 

This will include BitDefender , 

This will included Norton Antivirus 

 Sophos Av Software

Intego AntiVirus

 McAfee, 

Avast AntiVirus

Ad Guard,

 Webroot ,

 ESET ,

 Avira ,

AVG AntiVirus  

 avira antivirus 

Trustee  

AntiVirus for Mac

F-Secure

Securemac

Cylance

Kaspersky Internet Security

Total Virus Software 

VirusOne

Remove Virus One 

Read some of the posting and arrive at your own conclusions.