Gatekeeper take so long verifying files on my iMac's desktop

Gatekeeper is not supposed to repeatedly verify ordinary PDF or DOCX files simply because they are on your Desktop. ( Computer )

1 - Apple describes Gatekeeper as checking software such as apps, plug-ins and installer packages, particularly when downloaded software is opened.

Gatekeeper and runtime protection in macOS

2 - So I would not disable Gatekeeper to try to fix this. Your repeated “Verifying…” behaviour needs to be identified first.

3 - Please open Terminal and enter  xattr -l ~/Desktop/*.pdf 2>/dev/null

4 - Then  xattr -l ~/Desktop/*.docx 2>/dev/null

5 - Do not change or delete anything yet. If either command shows com.apple.quarantine

5 - ➡️ Tell me. If nothing is returned, tell me that too. ⬅️

6 - Also, please give me the exact wording of the verification window, wspecially whether it says “Verifying…” followed by the document's filename, or whether it names an application such as Word or Preview.

7 - That distinction matters, because Apple's documentation does not describe normal Gatekeeper behaviour as repeatedly verifying ordinary DOCX and PDF documents.

8 - One other thing. You are running macOS 15.7 and Apple has released later Sequoia updates, including 15.7.9, which contains security and other fixes. 

About the security content of macOS Sequoia 15.7.9

9 - I would assume that updating will fix your particular problem, but once we establish what is actually producing the verification, we will deal with that specifically.

Apple's relevant documentation:

Safely open apps on your Mac — Apple Support

Gatekeeper and runtime protection in macOS — Apple Platform Security

macOS Sequoia security updates — Apple Support

10 - Give me the Terminal result and exact “Verifying…” message, and I will continue from there without guessing.

I believe @Owl-53 is onto something there.

However, I recall another respected contributor @etresoft who knows much more about the inner workings of macOS (they are a third party app developer) mentioned something about the reason this occurs with files downloaded from the Internet, but cannot recall the specifics now (they posted about it sometime in the last year). I think it may have something to do with file type associations not being properly associated with an open known to be able to open them. I know I can open files with the "com.apple.quarantine" extended file attributes just fine on my Macs.

If I have an executable file with such an extended attribute, then I will be prohibited from running such executable by default. macOS is trying to be safe since those files with the "quarantine" attribute could be hiding an executable so only an app known to be associated with a particular file type such as PDF will be allowed to open such an app. If macOS didn't do this, then it could launch a compromised PDF as an executable which could infect your system.

You may want to try to change the default app used to open a problem file to see if that allows you to just double-click the file to open the document. If that works, then you can do the same thing, but next time click the button "Change All...." to cause all such file types to open by default with the same app. You can do this by selecting the problem file and selecting "Get Info". If associating the problem file with a third party app still has the problem, then try associating the file with the Apple Preview app as a test since maybe the issue is with the third party app you are using.

Choose an app to open a file on Mac - Apple Support

I really wish Apple would actually provide better documentation for this Apple security feature as it applies to documents downloaded from the Internet. Hopefully fixing the default app association will resolve the problem as it is the safest option if it works. If it does work, then each document type that has this issue would need to be re-associated with a default app. But if some third party software has corrupted the system, then who knows (typical culprits are anti-virus apps, cleaning & optimizer apps, and third party security software), but deleting a third party app that became the default app could have also broken the associations.

Have you tried to reboot the Mac to fix the verification loop

Gatekeeper and runtime protection in macOS - Apple Support (NL)

It is very possible my colleague @HWTech has also hit on something significant in regards to the Annoyance Pop Up

From one senior 2 year your junior, spend the time with grand daughter is more fulfilling than computer issues

Shall await your followup