I'm glad it was apparently comprehensible . I wasn't sure after I wrote it.
In general, with equal framing and focal ratio, you lose one stop of DOF when you go to FX. That because the focal length increases by 50% to keep it equal.
Comparing DX 200mm f/5.3 against FX 300mm f/5.6 you lose slightly over a stop.
There is not much you can do about it. We are not likely to see huge increases in sensor performance in the near future because sensor performance is quite close to the theoretical maximum.
Therefore the only solution is wider apertures. I run into this a lot, especially shooting groups of 6 or so performers at 200mm f/2.8, and especially on FX. I have to adjust my composition to either favor a single person or make the OOF subjects more acceptable.
You can also possibly just be patient and observant and try to shoot when your multiple subjects are in or near the same focus plane, combined with a suitable composition. You may want to talk to the play's choreographer to see if he/she can give you some better angles (just kidding ).
The better solution, of course, is to get closer and shoot wider where f/2.8 or so gives enough DOF but you can't do that.
Of course, if you could get closer and shoot wider you would then get sucked into even faster prime lenses like an 85/1.8 or 1.4, or the 105/2. So ultimately you can never get rid of the problem as long as you are always trying to mitigate noise. In the end you may need to select a happy medium of some sort.
You asked specifically about the FX vs DX example. I suspect you will see a dramatic difference on DX if you switch to an f/2.8 lens and shoot it there. Almost double what you see now.
For example, at DX 200mm f/5.3, 75 foot distance, your DOF is roughly +/- 4 feet but shrinks to about +/- 2 feet at f/2.8. Use a DOF calculator to get more specific and see other examples - you know the approximate range of distances involved and how people are typically spread out on the stage (hopefully).
Although in practice the f/2.8 lens has a total DOF of 4 feet, in order to take advantage of that you need a conveniently placed subject where the front DOF limit is reached, with everyone else in between that and the rear DOF, and one person conveniently dead center of the DOF to focus on. That does not always happen but it's something to try to look for in your compositions...
Otherwise if you concentrate on (and focus on) a primary subject of interest (like your daughter) who happens to be closest you then only have about 2 feet of DOF to whoever is behind her. You may have never thought about all that but fast lenses will force you to think it through while shooting.
When I am shooting groups at f/2.8, about 80% of my concentration and thought is on DOF issues and how I can seize a moment favorable to the DOF I need. If I am shooting one person that completely reverses.