Final cut doesn't open on Ventura

I am facing the exact same issue.

Is it resolved for you, if you can you let me know how you resolved it?

Are you using FCP 10.6.5?

"gives an error immediately upon startup"

What's the error?

"When I try to reinstall the system, I get only 2 shortcuts instead of 3 and, accordingly, I can’t start the process."

What does this mean? What shortcuts?

In just wanted to make sure the system is still running on 13.3.1 and not damaged by trying to install the older OS.

Does the application crash when you hold Opt-Cmd to start up FCP?

If yes, create a new admin user and log into that. Does the application still crash?

You tried it with a new admin user?

Please post an EtreCheck report https://etrecheck.com giving the application Full Disk Access in Privacy & Security System Settings. Copy the report and use the Add Text button to paste the result into reply box. 

Also try reinstalling FCP from the App Store.

That's not an error message. That's a crash log.

What OS are you running at the moment?

When you hold Opt-Cmd to launch does the application crash at start up?

I'm sorry I don't know where you mean. What preferences? You can't install an older OS on top of a newer version.

Yes me too same problem happening, didn't open

Do you also get an "Unexpectedly Quit" like the OP, or something else?