Thinking out loud again, and I don't have a working 5973 to play with-
I believe that when you load a method from Chemstation, it will get loaded into the MSD also and the display will show the currently active method.
Still, though, the whole system relies on one event completing and the next happening. When I've had a faulty or disconnected sync cable in the past, everything looks "normal" aside from the fact that one event doesn't trigger the next. Most recently I was dealing with this on a 5890 and the autosampler, where the run wouldn't start after injection. Still, though, I've seen MSs not start collecting data because they never received the "go" signal from the GC.
A few other thoughts:
You mentioned you suspected the detector in the MS. If it tunes correctly, this is highly, highly unlikely. Tuning, both the various auto tune algorithms as well as manual tune, is dependent on having a properly functioning electron mulitplier(the detector in this MS). You can look at the last tune report and see the tune voltage of the multiplier. If it's up near 3000V or so, it may be on its way out, but there again autotune would error out if it turns the EM voltage up all the way and still isn't seeing what it wants(specifically, if you watch autotune run, you'll see that it adjusts the EM voltage to get the target 69/219/502 ratios).