So it actually worked out of the box (looking at the topic title)?
And it worked following a factory reset.
(Which in the Android world doesn’t touch the installed OS.)
Which leads to the question: What were your exact steps towards this?
You can reinstall the OS manually …
Keeping the power button pressed for about 15 seconds should force a reboot in any situation.
Keeping Vol + pressed while starting or rebooting the phone should boot it into Recovery Mode.
Keeping Vol - pressed while starting or rebooting the phone should boot it into Fastboot Mode aka bootloader.
To work around touchscreen trouble when Android is running you can connect a USB mouse using a USB OTG adapter.
You might inquire here, perhaps … Fairphones in Australia 🇦🇺