There's a 4-way fitting off the primary tank that has the pressure switch. It closes at about 120 and opens about 90psi. It has an amphenol 3 pin connector (but only uses 2 contacts). You should test this switch. If the switch is bad the OEM ones are expensive but you can fit an HVAC switch with the same cut in and out pressures for a fraction of the cost.
The fact that you have no lights at all is interesting. You should have a blinking terrain light until the system has met the parameters for that selection. In other words it should blink, then the pressure switch will meet, then it will attempt to make the setting. Once the setting is made it will go solid. No light at all is not normal but perhaps there are other issues.
Do you have a black colored module or green? You appear to have blackout function to the CTIS module so I assume it's a black box that's the later ones.
The overspeed light blinking is good. Means the speed transmitter is working and the box is seeing it. Most of them are set to 55mph (when in HWY mode it's settable for each mode). When you go over that the yellow light on the dash will begin to blink. There is a time delay to activate. If you wish to change the settings you can use a Dearborn adapter, custom make a breakout cable (these trucks were never wired for the comms to the CTIS), and a laptop with free software from Dana. I've done this but it only works on the later black boxes with the J1939 protocol I haven't been able to link to the older green boxes. You can set overspeeds for each mode and inflation pressures.
I put a lot of work into my CTIS and developed tools to test hubs, talk to it and make field repairs if I blow a wheel hub. I've replaced 4 of the 6 hub seals it's a bit of a job but not crazy. A couple of things I like about it enough to keep it are the automatic checks so I know I'm good and the off road capability. My wrecker is quite heavy and when I get of road being able to air down keeps things from flying around. Also when I use outriggers for a lift I deploy them then hit sand mode and it sets right down on them. That thing weighs 37,000 pounds by itself it needs as much help as it can get.....
Lots of folks get fed up with it and ditch it but I wouldn't be without it. I suppose if I never knew any better I wouldn't care but a fully functional system is, for me anyways, a real feature of the truck.
Lots of threads here on the subject I've commented on a few. Do a search and read it all. Some of the stuff over in the FMTV area is similar. Those units are wired a bit different but essentially the same hardware. I'll be talking to my M1087 shortly the parts are on the bench waiting for me to have the time.