The switch does NOT fool the projector into thinking the lamp is there, it is to make sure the lamp door is in place. ALL projectors will attempt to power up when no lamp is present, but because there is no lamp, there is no feedback to the ballast control and the projector shuts down, giving the lamp error indication.
Projectors are designed for specific lamps and have very specific power and current specs and these specs can vary significantly enough between the different makes and models. And there is also the physical aspects, such as focal length of the bulb, etc.
As lamps age, they decay. And part of that decay can be for the lamp to become too weak to maintain the "fire". When the lamp goes out, the projector detects this and gives the lamp error indication.
This can also occur when trying to use so-called "compatible" lamps. What happens there is they can be right on the edge of being compatible with the power and current specs for the projector but after a few hours, they fall out of that spec envelope and no longer work.
It is something that is well-documented and very common. And it is something I have encountered many times in my years of maintaining projectors for local companies. That is not to say that the lamp ballast is not at fault, that can happen, as well. But I have seen many more bulb failures than ballast failures.
Unfortunately, there is no way to test other than installing a new lamp. You could try measuring the voltage, but you run the risk of severe electrical shock because most lamps fire on a very high voltage, then settle down to a running voltage. But without knowing what those voltages are supposed to be, you're still in the dark.