Ok, thanks for the Log - nothing unusual to see, except that you love AI aircraft, it seems - that shouldn´t have anything to do with it.
The ILS system in the 737-300 is really robust, no fancy FMS work necessary or so. Simply dial in the correct frequency and voila.
There are a few things that can throw people off:
Make sure that you have "disable tailwind ILS" deselected in the X-Plane preferences
Make sure you have not "failed" the receiver or transmitter in X-Plane´s failures
Make sure you are in range
the GS will be suppressed while you "head away" from the front course as set on the on-side CRS selector (LOC will always show, though)
loosing power to the corresponding radio
having an ILS frequency selected that is not corresponding to the frequency as maintained in X-Planes earth_nav_dat (often these are different from current day frequencies, as you can NOT update them with Navigraph or Aerosoft data)
I will attach two pics of my 737 at 16L in KSEA. Is this the same that you have set up?
Oh, and always great to meet someone that started back on the C-64 (Sublogic FS II!!) like I did .
Cheers, Jan