hellomf Is it possible to back-up the Pixel 7 to a computer and then transfer the back-up to the new phone (without needing to use the screen of the Pixel 7). Would this back-up include all app data?
There is no "cloning mode" that allows people who can't unlock the device to suck all the data out of it and put the data on another device -- for security reasons.
It may well be possible to connect a USB keyboard to the device and unlock it. And in theory it would be possible to operate some functions of the device without seeing the screen via a USB-connected keyboard and mouse, though it would be tricky. If there is already a SeedVault backup on the internal storage, it might be possible to copy it to an external storage device. Doing any of this would be very difficult without having another Pixel 7 running GrapheneOS sitting next to it to help simulate what the device with the broken screen would hopefully be displaying.
If the device were a Pixel 8 or newer, it might well be possible to use an external display. But for the older devices a working screen is pretty much a requirement for operating the device. If the cost of replacing the screen is more than the value of the data on the device, it may not make sense to replace the screen.