Regarding this problem, I have had a trouble free existence, so far. My understanding of the security on my car, an Elegance, is that I have a Red and a Green LED on the rear view mirror and the drivers door handle.
When I unlock, via the flip out one button key, the vacuum pump can be heard running, all doors open and the green LED flashes for a few seconds and the immobilize is disarmed. No hazard lights flash.
When I lock, via the flip out one button key, the vacuum pump can be heard running, the Red LED flashes for a few seconds and the immobilize is armed. No hazard lights flash.
If the immobilize isn’t disarmed, the Red and Green LEDs will flash alternate when the key is turned to position 2 and the car won’t start just as you describe in your first post… “not even a click“… As grober has mentioned, your central locking being faulty could be linked to your starting problem.
The above, is all the security that these 1994ish car had when the arrived at the dealers, however, additional security was fitted by most of the dealers in the form of a Scorpion alarm which came with its own 2 button fobs and a round key to turn off the alarm, as in your case, in the glove box.
On mine, the only functions that the Scorpion alarm fob carry out is to alarm the car, lock or unlock (hazard lights flash twice) the car via the central locking system, the flip out key immobilizer isn’t involved with the alarm and the alarm has no immobilizer which seems to be the case with yours too, so I can lock with either or both key fobs but I can only immobilize with the flip out key. Bear in mind, all these alarms can be wired in different ways, the set up in my car may be different to your car, it all depends on how the fitter wired it all up.
I think, as grober, that your problem is linked with the central locking.
There is a Comfort control unit in the boot, hidden down the side of the battery, held by two plastic nuts, one each end.
The vacuum pump is located behind trim on right, inside wing of boot in a foam box, it would be worth checking the wiring on these two items.
Also check the two 3v batteries in the flip our key fob.
Test the key fob
In the front fuse box, take a close look at fuse 27 and maybe 28 and 23
In the boot fuse box (to rear of battery) take a look at fuse 3
I don’t know if your set up will be the same as mine but these fuses refer to the infrared remote control, the flip out key works on infrared.
Comfort control unit.
Test the key fob.
Keys.
Fuses.
Dec