It could be, generally the burning position limitation comes from the way, how the lamp is optimized:
The hotter gasses are lighter, so tend to rise. That makes the arc bow up and all the upper parts becoming hotter than the lower, when otherwise equivalent. That could mean rather high temperature differences, e.g. the upper part of the arctube is attacking the upper temperature limit, but the lower remains rather cold, so prevents from the salts evaporating as you would like and in that way limit the lamp performance (efficacy, color quality,...). As with universal burn you do not know which part would be upper or lower, you have to keep all parts at safe distance, so they stay within the safe temperature limits in any burning position.
By restricting the burning position you know, which part would be upper and which lower (or which would never become the highest nor lowest), so you may shape the arc tube and thermal profile so, the differences become way smaller, so when you size it all so it still operates close to the upper limit (it could be even further than before, so usually that means longer life), the coldest spot becomes way warmer, hence the improved performance. You may see this in rather extreme form with the this example, where they were able to reach practically todays pulse-MH performance by the 70's probe start MH technology, just by reducing the temperature difference between the coldest and hottest point by making the arctube wall in constant disctance from the arc center.

Interesting how this works. The problem is what if it is installed and the lowest part of the bend faces down? Will the arc rise and heat the glass till it fails?

Exactly, it will get overheated and so fail soon. Maybe someone have already tried it and may describe exactly how it fails

In an environment with gravity field (or centrifugal forces) the heavier parts of the gas fills will follow the gravity and push out the hotter parts (plus the hot plasma is lighter by itself, so will tend to float upwards). As the arc is the heating element there, the gases from that will be pushed up, replaced from the bottom side by the colder ones. So we have classical upward thermal draft. And it is this gas motion, what blows the discharge stream into the known arc shape, so always upwards.
That means the hot plasma will float up, until the electric forces pulling it back straight become in balance with the buoyancy and draft, hence the arc shape, or when it reaches some barrier.
If the arctube is in designed position, the balance positiuon is in the middle of the arctube (that is the intention of it's shape), but if you will turn it upside down, the arc will just lean against the arctube wall above, so the short distance will make the quartz to severely overheat. And that means devitrification (it becomes milky and extreme fragile, till it explodes) if it is not that hot but long time, or just melt and blow if it is overheated too much.
Plus the way higher concentration of the metallic ions along the quartz will darken it as well.

I would think that a lamp like this would have a bayonet or specially designed base so you wouldn't screw the orientation up. That must be why we don't see them on shelves now.

They do, e.g. the Sylvania's use "Position Oriented Mogul", with a notch in the thread stopping the screw-in in the exact position...