Funny that "none" isn't one of your options, since on my Pixel 2 (android 11) it is. Of course I've not had any updates since November (3 years and that's your lot), so it may be that Google have removed it since then. Or maybe you have something else active that I don't that requires a lockscreen and so disabled this option (just speculating here, don't have any idea what that might be if it is even a thing).
There's no option within the phone for apps on the lockscreen (that type of stuff is manufacturer and android version dependent, so while it was a thing with HTC phones 8 years ago, for example, I've never seen it on a Pixel. And yes, it would ask for your PIN if you tried to leave that app). I've not used a third party lockscreen app for a long time, so don't know which if any will work properly. If you can't turn off your own lockscreen you would probably have to unlock twice with a third party app.
To be honest I rarely see the lockscreen: I put my finger on the scanner when I pick the phone up and it's unlocked. Therefore I don't find the lockscreen to be any inconvenience in use. Just mentioning this as an alternative workaround.