Given the historical nature of the modes, can you separate them from the temperament (or the tuning, if the comma was left unsplit)?
Tim, the person who knows a LOT about modes is Keystring, and her knowledge goes way WAY back. I don't touch things before the Renaissance.
But I think it is useful to do this:
1. Back to fairly ancient times, when the names and modes are nothing like what we know today.
2. What you are calling "Church Modes". When you start talking about tuning "back then", be aware that everything is based on text. We have no sound. Exactly how and when notes in a scale were adjusted from what we know as EQ is endlessly debated.
3. Modern use of modes, which is far more than jazz. They are all over the place in Debussy, Ravel, Bartok, and so on.
Re tuning, and you may sense my irritation merely because my ear for relative pitch is pretty freakish - which of course I can't prove here.
I started tuning pianos in my teens, so I'm infinitely familiar with the quirks of EQ. Coming from that I merely made note of how people shade notes in other systems, and it pretty much comes down to this. In diatonic music often 3rds are lowered different amount to a max of beatless. Then, using the key of C major, the same thing is done for FA, lowering the A, then GB, lowering the B, and 5ths are kept pretty pure for CG, GD and FC - for obvious reasons.
But these are all intellectual constructs, and the more you borrow from Peter, the more you have to pay Paul. If you have spent as much time as I have working with singers, you know that they are way more off on pitch than instrumentalists because OUR instruments to some extent guide us. There is no guide for the human voice except the ear. So the rules for keyboards are very different from the rules for other instruments merely because in the time of Bach and earlier some keys were unusable on keyboards. Never for singers, who are in practice often horrendously out of tune but POTENTIALLY better in tune than players other than string players, for the same reason - an infinity of possible pitches all equally easy to produce.
So in practical application musicians are much less reliable than they think in adjusting pitch, and this is why EQ is the system modern players start with. It means that potentially you, as a trombone player, only have to tune 12 notes, and all the players you play with do the same. Since very instrument in an orchestra has notes that differ from EQ, often in totally opposite directions, agreeing on EQ is a bit like everyone in a group agreeing to speak English, not because it is superior, but because it is common knowledge.
I won't go into all the out of tune notes on every brass instrument, and I won't even mention woodwind, but it's hard as h e l l to get several different instruments to tune together on one note or one interval in EQ. Yet until you can do that, you can't then adjust.
For advanced players you can then adjust for very diatonic and static music, agreeing how and when you will do it in whatever key you are playing. But you also have to understand that for an instrument like trumpet, if you are playing a high A, with 12, that note will NEVER be flat, always sharp, and there is no way you will play a beatless 3rd over F in that range. So if another instrument does that, there will be a unison off at least around 13 cents, and the result is awful.
In other words, on paper it's a wonderful idea. If you want to use an old system for tuning a keyboard, best for the key of C, and you write something in C, it will work. Modes in that key set will largely work because it's the same system. But when you move much away from the key of C, for instance to Eb, the whole thing collapses like a house of cards, and the key of Db will be just horrible.