Re: red gunk. I've only seen that once, on the PCB of a radio controlled car I had as a kid in the 90's! I vaguely remembered it flaked off like nail varnish.
As for that photo, going by the PCF8576T's datasheet, it looks like VLCD (pin 12) is connected to ground. This can be the case, as it only needs to be negative (with respect to ground) with larger multiplexing, so it looks like the whole thing just runs off 5V.
The "clip" that holds that flex cable in place looks like its still doing its job just fine. I would be very cautious about removing it to check for pad corrosion (a possibility on an old board!) in case you cannot get the flex cable and clip back on properly. But if you're confident, might be worth a check.
I have my doubts about the failed LCD driver chip ( the PCF8576). It isn't supplying any significant current anywhere, so will never be stressed, but I really couldn't say for sure - I agree that checking its outputs would be worth checking just in case, a good check would be seeing if the outputs reach close to the power rails (0v and 5v). My vote is the glass itself. Excessive IR or UV from sunlight can age polarizers, and I have had several old watches "fade". I don't know if this is the liquid crystal or the polarizers.