How can I prevent the annoying password screen after inactivity?

Ok since you have tried all 'require password' settings you should list them.

did you check System Preferences> Security & Privacy> General> Advanced

make sure Logout after xx minutes of inactivity.

It could be a corrupt preference file.

Click on Finder, click on Go To Folder enter the following,

~/Library/Preferences/com.apple.systempreferences.plist

press Go

A new Finder window will open with the file highlighted, move it to the Trash.

Don't empty the Trash yet. Restart your Mac. Open System Preferences and set

all your require password settings again. Restart your mac. Now run your mac

until it sleeps or the screensaver starts, give it some time then see if you still need a

password to gain access to the mac.

About the bug report, no point telling us, you are just in touch with other users.

If there was a real problem then you would see a lot more references to it in these communities.

System Preferences> Security & Privacy> General make sure Require password .............. after sleep or screensaver is unchecked.

Or if you don't want the mac or the screen to sleep then System Preferences> Energy Saver and set either one of the options or both to never.

So when you moved the preference file to the trash and then restarted your mac did you then open System Preferences and set your options that no password is required on waking from sleep or screensaver and then restarted your mac. And then test to see if it will allow you to wake without passwords.

Have you tried starting in Safe Mode and then rebooting.

Also try setting up another user account (Administrator), make sure settings are for no password required, and test again, this should determine whether the problem is global or just in your primary user account.

Do you have any third party apps installed that could influence the settings on your mac.

My install of Catalina wakes from sleep or screensaver without the need for passwords.

Boot into Safe Mode (Use Safe Mode to isolate issues with your Mac and Playing Safe: what does Safe Mode do?) and try sleeping and waking from there. Reboot normally and test again.

NOTE: Safe Mode boot can take up to 10 minutes as it's doing some system cache cleaning, volume verifying and directory repairing. 

My wife's new iMac did the same thing as you reported. We also had all options and preferences set correctly. She had done nothing to cause it and had changed nothing. It easily cleared out when I did a safe restart which apparently cleaned out something. Now everything is back to normal. I hope it works for you as well.

What do you mean it takes a few days to rear its ugly head.

Are you not able to have your mac running and change its settings so that it either goes to sleep in 5 minutes, or the screensaver starts in 5 minutes, surely that is easy to set up and experiment with.