The problem is, what do you do at 24:39:35, when both the minute and the second arm have more than a third of the way to go before zero?

Any mechanical design where units are not cleanly divisible would have to devise a way for the arms to "fast forward" over the gap and accept nonsensical results for a few seconds before midnight.

Like I said, the arm keeps moving the way it already was, it doesn't jump forward because time isn't jumping forward. You shift the position of zero over. So rather than zero being at the top of the clock, the minutes marks spin about a third of a rotation and now where the minute hand is pointing is zero. It's no different from having a "day of the week" dial on a watch which jumps forward one tick every day.

It seems to me that if you rotate the minutes dial in a quick motion before midnight, you have both the "nonsensical times" problem for a few seconds and also a massive usability problem throughout the day, since the minutes mark are not where people expect them, the minute numbering is upside down or nonexistent, and the visual cues of vertical = half hour intervals, horizontal = quarters are all gone, killing the one of the killer features of the analog clock.

I can't see how that solution is superior to fast forwarding just the much lighter and already mobile minutes arm, but perhaps I'm missing something.

Well it really comes down to what you are trying to represent - moving the minute hand works well if you are only concerned with subdivisions of the hour, but if you are trying to keep track of minutes, you want to know how many minutes have passed between one point and another, so for example if I am waiting for something to cook for 45 minutes, I want to wait for the minute hand to make 3/4 of a full rotation. If the minute hand jumps a significant amount, then that no longer holds true. By moving the marks instead of the hand, you keep track of the passing of minutes naturally.

I don't personally agree that it would be a usability problem - people already use wrist watches which aren't perpetually oriented up, they naturally find the zero point and understand the position of the hands relative to it. The only difference is that the zero point for the minutes will not generally be the same as the zero point for the hours. I certainly don't buy the "it's not the UI we're used to so it is a worse UI" argument. Mechanically, incrementing the marks is much simpler - I'm not even sure how you would get jumping the minute hand forward to work normally nonetheless how you would handle things like adjustments; but for moving the marks it's just another gear, which most clocks would have anyways for incrementing days. I don't think reading the time for the brief moment when the marks are in motion (and thus you know it's precisely martian-midnight) is really any issue; the only edge case I can imagine is a back to the future type situation where you are trying to determine a specific time by looking at a stopped clock (or more likely a still photo of a clock) and can't see that it is in motion, but the confusion would go away immediately when you see the hour hand pointed at the zero position.

To "jump forward" you can just stand: once you reach 24:39:00 the seconds arm stops over zero, waiting for another 35.2 seconds; at 21 seconds before midnight the minutes arm gets picked up by the seconds gear and advances one display minute per second of time, until midnight is reached (thus readable as a "seconds to midnight" movement in settings such as New Years). I have seen more complex motions implemented in wrist-watches so it seems doable.

I'm not yet entirely sold on the moving dial principle, I've placed my wristwatch upside down on my hand and trying to read it gave me a headache. The angle the hands make with the line of sight is indeed not very relevant, but their relative position to the fixed known elements of the dial feels essential.