If I got you right, you can see your Pixel's screen without issues. It's just its touch functionality what doesn't work fine, right?
I've never tried myself what you need to do. Here are my raw thoughts on how things would work and what obstacles you may find during the process:
I guess now you have your Pixel's original OS installed (i.e. Google's). I think you should be able to enable the developer settings and allow the bootloader to be unlocked, just with a keyboard/mouse. So no obvious issue here.
In fastboot mode, external USB keyboards and mice won't work, but you should be able to unlock the bootloader just by pressing your Pixel buttons. Then, GOS flashing could be performed. So no obvious issues here.
GrapheneOS has restrictions for the USB-C port, as an exploit protection. It's configured by default as "charging-only when locked." So, with GOS default configuration, you won't be able to unlock your device after a reboot, cause you cannot enter your password/pin with your keyboard/mouse (they won't work until you unlock your device). You can of course change the USB-C configuration and set it to "On". With this setting, you should be able to unlock your Pixel without issues. So, there is an issue here but that can be surmounted if you change the default configuration.
One more potential issue, related to the previous one... When you install GOS and start it for the first time, the first-boot wizard appears, as with any Android system. I don't know if this wizard stage would be considered as an "unlock" as for the USB port . If it's not, then you won't be able to proceed through the wizard cause your USB mouse/keyboard won't function. In this case, I think it would be game over and you won't be able to finish the installation of GOS. If it's consider an "unlock", then you should be able to finish the wizard, go to the settings and configure the USB-C port as "always on". You should change this setting as soon as you finish the wizard. Otherwise, you won't be able to unlock your Pixel if you reboot it. So, in summary, there is a potentially unsurmountable issue here (it could work or not, you would need to actually test it).
Those are the most obvious issues I can think of. Let's see if someone else wishes to add any more potential issues that I've missed.
Anyway, I suggest you try to do it, to see if it can be done or not. GOS' web installer is really easy to use, even for users with no previous experience flashing Android devices. It won't take you a lot of time and you'll know for sure.
Between GrapheneOS and iOS, I strongly suggest you give GOS a try. It would provide you much better privacy than iOS.
As for compatibility, most Android apps run without issues on GOS. The only problematic ones are some banking apps or the ones that use the SafetyNet/Play Integrity API. (This is not an GOS-exclusive issue, the same thing happens with any alternative Android OS that is not the one supplied by the device manufacturer).