The PCjr does very much have a ROM for the character generator. It's basically the same as a CGA card, but with some more graphics modes, circuitry for sharing the memory with the main system, and an ASIC is used to implement it.
It also looks like the addressing to the character ROM has failed. Try to boot up a game (something that is pure graphics) to see if the fault continues there. If the game looks normal then it's the character ROM or the addressing logic to it that needs attention. (I think that in the graphics modes the text does not come from that ROM and comes from a bitmap in the BIOS ROM instead, so text displayed in a graphics mode is a good test too.)