From your description, I get a rough idea of what your problem is. The external HDDs that are slow to mount on Mac may be due to Time Machine, system issues, software conflicts, file system errors, etc. Here are my suggestions to fix the problem. Hope this helps you out.
1. Bypass Time Machine verification
You say that you use these hard drives for file backups, so what you see in Activity Monitor about the two drives doesn't mean they have any problems. It probably indicates Time Machine is verifying your external hard drives before start backing up files. You can click on the Time Machine icon in the Apple menu bar and select "Skip verification".
2. Boot Mac into Safe Mode and then connect the two drives
If that makes no changes, test the performance of the two drives in Mac Safe Mode. This is to exclude the recently installed software conflicts.
3. Connect the drives to another Mac
And you are also suggested to connect the two HDDs to another Mac that's not running macOS 13 to verify if it is the macOS Ventura problem causing the external hard drive not mounting on Mac.
4. Run First Aid to check the two HDDs
The external hard drive that takes a long time to mount on Mac can also result from file system errors. You can use Disk Utility First Aid to check and repair errors on your external hard drives.
To run Mac First Aid to repair the disk, you need to:
- Open "Disk Utility" in the Applications folder.
- Select the drive on the left sidebar.
- Click "First Aid" on the top.
- Wait for First Aid to complete checking.