happy wanderer
Day Dreaming ...
The power for the light can come from anywhere, it does NOT have to be the bike. A separate battery eases the load on the /5 electrics and doesn't introduce another variable.
I have seen first-hand a couple of times where the strobe was not as constant or regular as it should have been - which may have caused the effect of "not as bright" - and that did point directly to the issue in those cases. This was with a Craftsman inductive light, which I still have in my garage but haven't needed for a while...
I realize the power source is not relevant and in fact, sometimes it is better to use a 12VDC source that is not connected to the bike being tested.
The point I was getting at is that I think the intensity of the light itself changing is likely due to missed pulses i.e. coils not firing, bad wire, worn out resistor cap, bad plug. The misfires could create that illusion of less light intensity. So I do not think that the timing light (when it does fire) is any less or more intense though, it just looks that way.
And yes, when the bike is missing or misfiring you definitely see the evidence on the timing plate on the flywheel. It disappears, jumps and generally is just not stable. Your advice to move the light side to side is a great way to figure out if it is only one side. This usually tells me to look closer at plugs, wires and caps. If it is both sides, the common components are now the coil, ICU and bean can including the connectors and wiring for those parts.