Changing the finger for repeated notes is a bit controversial, and I think it largely used to be a convention in the early 20th century. I think it often produces slightly more legato touch, but not always. I think it is something you should be comfortable doing.
I'm going to leave your other points, which seem more directly suited to the OP, and are very well articulated. That's better left to the OP and you.
But this one intrigues me a bit, even if it is a bit going a bit off topic: I've never actually heard that single-note trills using multiple fingers is controversial.
There is some historical precedent that suggests the "hummingbird"-style repeated "pecking" at the key with one single finger worked well for pre-modern pianos and harpisochords, and anecdotal evidence that some modern jazz organists like Jimmy Smith, Joey D., and others use the single finger technique. I don't necessarily believe it in the case of Hammond organ, despite having had the chance to sit behind a number of the living and recently deceased masters, as an observer/regular audience member. Perhaps I just didn't notice.
It's very quick, after all, and doesn't need much histrionics to execute.
For jazz, and on the Hammond, while it's very common, it is an ad hoc kind of thing done while improvising. I haven't seen, or done, any specific patterns (as in, I haven't explicitly practiced this technique, but rather I just use it): it depends on where on is, physically, in relation to the manual. It could be a 12 trill in RH, or 321 , or 432: the keys are physically different on organ, with no "lip" where the top of the ivory is on a piano. They call them "waterfall keys" (at least on the Hammond organ, and probably some other electro-mechanical organs...I don't think this is a standard design for traditional organs): mostly it's helpful for performing glissandi without cutting one's fingers/hands, but it does make repeating trills marginally easier by letting the fingertips slide off the front edge, without risk of catching the "lip" of the key as one returns from having depressed the key. So, even an habitual "thumb-avoider" like me can still make a 12 or 123 single note trill in the RH, sort of similar to how one would in the LH at the piano keyboard, doing a "regular" trill.
It's just a flourish, really.
I think it's without question that on a modern piano, or at least a fascimile thereof, using a few fingers rather than just one is more effective.
Easier, really. For me, like I said, I have a hard time using the thumb, although it's possible (and I've done it in the past and likely will many more times), but even with just 432 in the RH, I get good speed and keep the fingers just barely above the keyboard. IOW, it's a compact and efficient technique.