There's no single app which can restore convenience after MS Phone Link removed the GUI emulation for PIN input (thanks to Tim Apple), I guess nobody cares about the niche of a dozen people who want to remote into their own phone. Rustdesk or Teamviewer can get to the blind PIN screen quite easily, but not when the phone was locked for a minute. Phone Link actually stays awake even when the phone is asleep (latest version on the phone and on Windows 11 as of today, S22 Ultra, Android 14, One UI 6.0) because you can always access messages, photos etc. without waking it up. But it can't nudge the phone out of the sleep mode - however MyPhoneExplorer can (that's a better version of Phone Link which supports file transfer over Wifi). So you can wake the phone up with MPE and hope the remote control's service comes back top life...
A more viable method is remote unlock over ADB - you can send some simple commands to wake up the phone and send the PIN as key inputs. If your phone supports wireless debugging, you don't need a USB connection and just run a simple batch script to unlock the phone and then connect with Phone Link or any other remote software. I think it has to be reenabled after reboot though. Of course you have to give your remote app projecting permissions (for Phone Link: ./adb shell appops set com.microsoft.appmanager PROJECT_MEDIA allow) but that's trivial. Teamviewer paid Samsung an ungodly amount of money to circumvent the projection permission via Knox (there's some trivia for you), but it still can't show a "secure" screen or keep the host service alive when the phone goes to sleep.
Getting ADB to work should be very simple - enable developer options and USB debugging and use the latest platform_tools from google. On first connect you'll get a popup on the phone where you have to whitelist your PC. What does it say when you type in "adb devices"?