The PSU is now fixed and the display is bright, clear and ghost-free now
As well as the dead zener, one of the PSU caps (a 6800uF 35V Nichicon) had sh*t itself too. Good job I was going to replace it anyway.
I did a final check of the whole unit with a thermal camera before putting the lid back on, and found a tiny resistor getting very hot whenever the unit had a tape in. I didn't recall seeing that during my 'before' check, so I had another look at the service manual...
This resistor (R380, for anyone wanting to take a look) is part of a discrete transistor circuit that drives the mechanism. All the current that flows through it ends up in the base of a small transistor, so there's no way it needs to be passing enough current to get it hot... is there?
The value of the resistor is 330R, and sees the whole of the "+10V" supply across it normally, which would result in a dissipation of 300mW. However, the 10V rail is unregulated and comes straight from the mains transformer, and since this is an old EU model that I'm using in the UK (240V vs 220V), its voltage is significantly higher - about 13.6V. This means about twice the designed-for dissipation, and a glowing white spot on my thermal image.
(It's exactly the same in the BioLogic unit too, so I'm fairly sure it's a design feature rather than another fault as such).
It looks to be part of a simple logic circuit that turns a solenoid on and off, and I can't for the life of me think of any reason why it needs to pass so much current into the base of a high gain transistor, so I've swapped it for 1k8. Now it dissipates 100mW and is warm, but not worryingly so. I could probably increase the value further without issue.
I also checked it against the PCM-2300 service manual. In this version, there's a little '!' warning next to it on the schematic, but the value is the same. Maybe someone at Sony spotted that it got hot, but didn't have the authority to actually change it?
Either way, with the PSU fixed and R380 changed, the unit is back together and working perfectly. Result