If I'm understanding your description correctly, the tracks and the opener are mounted to different joists and the joists are prependicular to the tracks and opener? Think about the forces applied to the door in order to open it... the door operator is mounted to the header over the garage door. That's the surface that takes the stress of the forces applied in order to open the door. If you dont believe it, disconnect the opener from whatever its hanging on, hold it up there in your hands and operate the door...there is no force applied to the ceiling other than the weight of the opener itself.
The weight of the door, when it is open is indeed bearing on the track which in turn transmits the force to the track hanging hardware and to the associated ceiling joist. Again, there are no forces applied to the track hanging hardware in a direction parallel to the door travel or laterally. It's all gravity.
While the weight of the opener hanging on a joist certainly will apply forces in the direction of the observed sag, The majority of your deflection is no doubt simply the result of a really long piece of wood spanning 20 odd feet for fifty years.
The weight of the door is putting much more weight on the other joist than the opener is putting on its associated joist but the door tracks are closer to the exterior walls of the building...closer to the point where the joist transfers its weight to the wall. You can hang a lot more weight in that case than you can at the mid point if the joist.
In any event, your least expensive and quick way to stabilize things would be to tie several joists together with steel angle lagged into the top surfaces of the ceiling joists, at least one bay ahead of and one bay beyond the joist with the weight applied to it. this will distribute the weight over 3 joists in that case. Don't expect this to remove the sag that's already there...it won't go away even if you took all the hardware off the ceiling, too much time has passed on those poor joists. If you do want to make the ceiling flat, consider sistering a beefier framing member along side the joists. You would need to jack the existing joist until it was "crowned", attach it to the sister joist which should be installed crown up and when you take away the jack, the forces will be transmitted to the sister joist and if you're lucky, it should flatten out. Admittedly it's kinda a **** shoot because you're taking a SWAG at how much you need to jack the existing joist and how much the new joist will sag just of its own weight givein the span.
Tying the ceiling joists into the roof rafters may help also, especially if the roof sheathing is 1x boards as opposed to plywood. The mechanism that comes into play here is two-fold. The weight of the ceiling is now assumed by the ceiling joist and the rafter. The good thing is that the rafter, no matter what its dimension is a little more than half as long as the ceiling joist and the force you are adding to it is not being applied perpendicular to it. Couple that with the fact that any deflection of the rafter is somewhat transferred to the adjacent rafters by the sheathing and you may find this to be an acceptable remedy.
I know this is kinda wordy but I hope it helps.
