That ripple reading is reasonable so I think we can rule out ripple.
I'm pretty well stuck. If Q1 (the IRF840) were getting hot I could understand it. As Ian has explained elsewhere, when the clock is in the initial start mode (scrolling digits) Q1 is under some heavy stress. But that does not explain IC1 (the 7805) getting hot because the HV circuit is fed from before IC1 not after it.
I don't know how far you want to go with getting meter readings. We've covered all the relatively easy stuff that I can think of. It would be interesting to know how much current (milliamps) is being drawn from Pin 3 of IC1 but to check that you'd need to disconnect pin 3 from the board so as to insert your meter in series (between pin 3 and the hole pin 3 goes into). This probably means pulling IC1 out of the board, bending pin 3 out to the side, soldering a wire into the hole pin 3 goes in, putting IC1 back in the board with only pins 1 & 2 connected, connecting milliamp meter between pin 3 and the wire, etc. If it was my clock I'd do it, but I've been soldering etc. for 30+ years so to me it's no big deal. Not knowing your level of experience I don't know it you should try this or not... Up to you. No guarantees this would solve anything anyway but it would be interesting to know.
Look into it later when the dust is clearing off the crater.
Pulled IC3, and IC1 does not get hot anymore. Factory reset again, IC1 takes longer to get hot, but after 30-45 seconds of the digits cycling they go blank, then display a number, which I assume is the version number, but the tick LED starts to stay on, and fade out during this. I unplugged at that point, and decided to post.
It's very strange. I have been studying the thread and can't make much sense out if it. At this point, I think the return/replace option is the best bet. With really weird ones like this, I like to do a post mortem on them to improve the design. Would that work for you?