I'd verify that your starter relay is closing and staying closed when the button is pushed, it should not open unless the door is opened or the machine is otherwise stopped. It is a true relay, not a momentary switch like other machines have. These fail often enough that I carry one on my truck.

Do you have 120 at both ends of thermal fuse (or 0 potential across it) when the machine is trying to run? This is by far the most common reason a dryer will not run.

After that your timer could be losing the neutral (dropping it we'd say).

The elec control handles moisture sensing, I'd ignore it unless you have other reasons to suspect it, such as blatantly burnt resistors.

I find that most diagnosis is far simpler than it seems and I tend to go with what 10+ years of mistakes has taught me. It's difficult to convey how I operate and I've become over-confident with live work over the years. I don't want to lead you into a process that could harm you.

Neutral and hot meet at that start relay and I've changed many. This site allows returns I think so you might order start switch and timer and hope. You can test that live with little risk. When it is operating normally (or should be ) there is 0VAC potential between light blues. Meaning that there is no voltage difference(potential) between those terminals and therefore they are electrically connected. So with your meter on AC voltage you should see 0.

The motor would need 120v hot to neutral to run, so if the timer is not providing a good neutral you'd have trouble getting the motor to start. If you're seeing voltage #s like 45v something is odd, maybe a burnt contact in the timer or start switch.
You should still get a reading on the motor windings even unplugged at standstill. In the lower corner of that diagram is the motor plug layout. You can check the start and run windings using the #s indicated. The winding resistance is fixed, regardless of motor status. The only thing that changes during run is that the centrifugal switch closes and supplies 120 to the element, completing the 240v needed there to heat.

One thing that we've all missed at some point was a flipped breaker, bad cord, or bad outlet. It's worth verifying that the machine is receiving 240 or close to it at the terminal block. I've seen plenty of bad terminal blocks after hours of frustrating diagnosis.

There are very few reasons a machine will not start, Bad timer, bad start switch or relay, insufficient power supply, and thermal fuse are the main ones. Thermal fuse is 75% of the issues. You may have one of the 1% of odd ones.

You have a fun one! Only runs during wrinkle prevent...

If I was called to your house I'd get you a timer with fingers crossed. I still say ignore the moisture sensor board, as I rarely see them fail. The fact that it runs at all means the thermal fuse is good.

If you are comfortable with working on a live circuit you can pull the drum out and reconnect the front panel off to the side so the door switch is connected. Most of the live stuff is behind the bulkhead in this dryer. There's just the motor in front. Still use nonconductive gloves and eye protection in case of a surprise... I've both welded and vaporized meter leads over the years. As a general rule we don't work live unless there's no other way and I don't like even advising you to.

There are several 120v lines at the motor connector, a neutral, and a ground. When it should be running you should see close to 0v or 120v between any 2 terminals in the plug, my guess is you'll find an odd # when you test to neutral at the motor connector. We know the motor did not fix it so it's between the timer and the motor. If wrinkle prevent works you can use that to ID where you should be seeing 120v on other cycles. While you're there inspect the actual motor terminals on the wire harness for corrosion or burns.

If you have an air dry setting use it. If something changes and the motor runs on a heat cycle it will also be supplying the other 120v to the element and you can't let it heat for long while open like that.

They're dumb machines and quite simple so I am annoyed when I don't quickly find the issue. In this case process of elimination and experience says timer but doing 30-45 calls a week I am bound to be wrong once in awhile, also just part of the job.

Good luck, I'd like to hear what you find.