The "circle with a line through it" is known as the prohibitory symbol. Typically this means that your startup disk contains a version or build of macOS that your Mac can't use. In your case it seems something else is going wrong, but, if I read between the lines correctly, it seems your main goal is to upgrade to macOS Big Sur anyway.
The best course of action is really to make a full current backup of your system and then reformat your drive before re-installing macOS.
You have a number of options open to you, such as using the Option-Command-R keyboard shortcut at startup to upgrade to the latest macOS that is officially compatible with your Mac (and assuming macOS Big Sur is compatible, then this will be what is installed).*
If using this option you'll have the ability to reformat your drive using Disk Utility first before opting for the Reinstall MacOS option. More information on using macOS Recovery to reinstall the Mac operating system is available on Apple's website: How to reinstall macOS.
Another option is to just create a bootable installer for macOS and then boot up from that to install macOS Big Sur. You can opt to fully reformat your drive before doing the installation. More information is available from Apple's website: How to create a bootable installer for macOS.
Regardless of how you go, the main thing here is that you're broadly following this order:
- make a full and current backup of your system
- reformat your drive
- reinstall macOS
- transfer across your files etc
Following this order means whatever is causing your 'wonky' Catalina installation really shouldn't come across with you.
Keep in mind that many users have had issues with macOS Big Sur, so whether you should make the jump yet is really only a decision you can make.
* If your Mac has the Apple T2 Security Chip and you've never installed a macOS update, this will install the latest version of macOS previously installed on your Mac.