I’ve been wondering for a while now why no one has ever tried using a magnetic arc deflector with a metal halide lamp. They were used with medium pressure mercury lamps to prevent the arc tube from melting when the lamp was run horizontally, so why aren’t they used with metal halide lamps run horizontally to try to prevent the arc tube or the outer bulb from melting or bulging, especially with tubular lamps?

I don't know the exact reason, but it might have something to do with Curie points and magnet prices.

The arctube is probably too small and it would require a very precise magnetic field. This would be hard to get through the side of a lamp.

There is no need in precision - nothing happens if you apply field over larger area than the arctube. You will need bigger coil and core to generate the extra field tho

The main reasns are probably cost and efficiency (losses of the deflector). Besides, maybe different halide lamp models (different chemistries, different manufacturers etc) have different strength of convection, so would require different force to keep the arc centered

As said above, because it is cheaper and maybe even more efficient to just design the lamps to handle the load.
With MA, the engineers were "in a corner", they pushed the tube material to its limits (and had no other material available - with quartz they were not able to make reliable seals) and the tube was quite long (so the bowing would be very significant), so they did not have any other option.
Plus the deflector is rather heavy and can not be elsewhere than on top of the lamp, mainly with remote ballast stadium fixtures it would mean both weight, as well as position limitation problems (those are aimed at different angles, so each fixture would need the deflector set specifically for the angle, plus it would obscure the light)

Now the quartz seals are rather common place, the tubes are way shorter, so it is just a matter of design (the cost in efficacy is few percent, way less than would be the deflector losses; plus dont forget the deflector are structures obscuring the light and it would be definitely cheaper than the deflector).

I have also been wondering about whether using a magnetic arc deflector would help to extend the life of universal burning, base up burning, and base down burning metal halide lamps if they were operated in a horizontal position such as in the arc deflector variant of the GE Form 109 lantern.

Again, maybe yes, but still it is cheaper to design more robust lamps or just accept the a bit shorter lamp life.
Either way cost some money, but the deflector is still one of the most expensive options.
Otherwise it would be simply used as a standard thing...

As Max has said on this picture the lamp chemistry has improved over the years so they aren't as sensitive to burning position.

In fact early metal halide lamps did use something like a deflector wire construction in the bulb. The Osram HQI-T (Radium HRI-T) 2000W/N/I from the 1970s and early 80s did have the mounting frame which acted like a stabilizer to center the arc in the rather long burner.

The detailed discussion can be seen here:
https://www.lighting-gallery.net/gallery/displayimage.php?album=1488&pos=249&pid=30803

@Monkeyface: But that mean those lamps would have to have their (horizontal) operating position restricted to the wire being above the arctube (so pushing the arc away from the wire would be pushing it away from the wall as well).
Can not imagine this limitation with a screw base lamp, so I doubt it was really meant to work that way.
Plus field from just a single conductor pass would be too weak to do anythingto the arc, I'm affraid. The MA lamps had to use a current transformer to feed few 10's A through the deflection compensator conductor (or the deflection compensator had to use iron core magnetic cirvuit with many turns on it to get the required effect)...

So to me it look more like some magnetic deflection may be observable, but the lamp design itself does not rely on it in any way.

Michael is correct, that the original HQI-T lamps were indeed designed with magnetic deflection.
More recently it has been applied in other lamps as well.  Perhaps one of the most famous was in the USA Philips CMH Retro-White lamps of 250W and 400W in which they reverted from the usual half-frame wires alongside one side of the arc tube, to a full frame construction with current loop so as to exert Lorentz forces on the arc and improve stability.  For those kinds of long-arc ceramic metal halide lamps the effects of gravity can be quite significant, and the change of mount frame delivered a notable improvement in lamp performance and life.  Osram applied the same trick more recently in some of its photo-optic projection HMI lamps.