Well, sounds like shift cam dial pin channel dirty or maybe axle shafts/gears need cleaning.
Not sure if it would help, but may remove shift cam stopper and spring from angled drain plug, then try shifting in place by rocking.

Maybe stick a borescope in the view port near top of clutch in R crankcase. Of course, this means removal of clutch.

If your motor is a "74 or later i'm guessing the gear change shaft has moved sideways while the cover was off then gone back one tooth out rather than central.www.partzilla.com/catalog/yamaha/motorcy...fter-2-250b-c-400b-c
The motors for sale one is a "73 250 at best & the 360 looks an ordinary piston port RT1 of say "70, before "72 anyway. Has the std twin plug hole angled plug head not center plug as the piston port MX had. That's my take on it. Be good to have if you had the older model frame up to "73.
Imgur pics wont come up on computer i'm on but will try later on other one. Great that you got the clutch working ok.

I’ve acquired strips of Velcro and now tie back all the clutch levers on my bikes when they are parked. I haven’t had to do the rolling start on any of them since.

Well, bad news today.
I started the bike so that I could figure out the shifting issues. It started and idled for a second, then I heard a loud metallic clank and saw a shard of metal fly away from the engine. It turns out that one of the screws that hold the flywheel magnets stripped out of the magnet and destroyed the flywheel cover. Now, one of the magnets is dragging against the coils. I think I would probably have to replace both the flywheel and flywheel cover.
Here are some photos of the carnage: imgur.com/a/xGnCWKO
At this point, I don't know if I want to continue with this project. I don't really have the budget or time to fix all of the many issues that this bike has.
What's everyone's opinion? I think that this motorcycle needs a lot more attention than I can give it right now.

Blimey, you're having all the luck [bad luck] & sorry to read that.
Very minor point also if you 'do' carry on but the original flywheel proper nut is about double length of a normal nut.
I guess it's like that for a reason with the long thread to reduce the chance of stripping the nut when tightening super tight.
Is the magnet held up by a plate. Is the plate still usable--threaded hole etc.
Can't believe the luck you're having.
If the magnet broke in haf then that's another problem as must be replaced or mucks up its noth south--i dunno but something not good.
I've never had that happen but did loose a flywheel once when all the rivets broke in the middle on my 360.
Actualy that reminds me one of my kids broke a crank at the left main bearing when we were roadracing 100cc at the airforce base with no cover to let it cool as much as possible & it took off at about 10,000rpm & never found it--flywheel & shaft.
Went into the refueling big tanks area at the local airforce base.
Couldn't help thinking hell that would've made a huge explosion seen from outer space if it went off.
Might've been the same day [annual 30 lap GP] that i broke a crankpin in half.
Was a pin from a Suzuki that pumped 2stroke oil through a hole in the crankpin & couldn't stand the revs i guess.
Jeez did it vibrate all of a sudden lol. Not a good day 'that'.

Thanks for the words of encouragement, RT325.
I slept on it last night and things look a little bit better this morning. I think I should be able to tap out the magnet and install a larger screw, I don't think the magnet is split (hopefully). You are also right about the flywheel nut. The brilliant previous owner of this bike swapped out a lot of the hardware with incorrect stuff. Just the other day, I was trying to get the rear brake lever off and the bolt holding it in place snapped in half. Turns out it wasn't even the right thread pitch.
I think what I will probably do is try to epoxy the old flywheel cover back together and hope for the best. I think the cover is magnesium, can anyone confirm? The place where I am working now has an aluminum MIG welder so I could potentially weld it if it was aluminum.
I will see how far I take this project. I have until early August to finish it, because I will be moving to an apartment that frowns upon people working on stuff in the parking lot. Hope to get pretty far by then!

Another small update this morning;
I took the flywheel off, and one of the magnets is cracked. I don't know if I can just remove the broken piece and reattach the magnet, it would probably do something weird to the timing. There's no harm in trying at least.
Broken piece: imgur.com/a/Sad9wTh

Are you all ready to see the worst metal fab repair you've ever seen?
I riveted two pieces of aircraft aluminum to the cover to close the cracks and seal out water and debris. There's also a bit of JB weld holding the piece on that actually flew off. It's terrible, it looks ugly, but it does what it needs to. I plan on hitting it with some black spray paint so that the sketchiness is less noticeable. At the very least, it matches the aesthetic of the rest of the bike. imgur.com/a/LZGdTHU

There is some good news, though. The bike runs just fine, the cracked magnet doesn't seem to affect the way it runs at all. I replaced both screws on the magnet, coated the threads in blue loctite, and used my pneumatic rivet gun to make sure that the screws do not back out. I guess I'm going to have to dig in to the transmission issue soon! Maybe when it's not so hot outside.

The cover obviously is no barrier to the shrapnel. I think it was more to protect the electronics from rain, mud, etc.
I would consider a new setup. I believe VAPE might have a basic kit just for ignition. Of course, it has no key way, a disaster also waiting to happen.

Flywheel key is more of an assembly aid...  the taper provides almost all of the holding power.  Machines like drills and milling machines transmit a lot of power to the tooling with just a taper, no key.
If you have a good clean taper fit and no key, no problem.
If you have a poor taper fit with a key, the key will shear very quickly.