ON a 1993 NSX the security system is armed by locking the driver or passenger door from the outside using the key.
As noted, the security control unit is located behind the glovebox so you need to remove the glovebox as described in the service manual to get access to the terminal plugs. As I recall the SCU is the bottom module in a stack of modules at that location. I have never removed the module and to do that you might have to remove more than just the glovebox.
Four things to check before you start tearing stuff apart:
1) Was there ever any work done on the driver's side door? If so, during reassembly the plug that connects to the indicator light may not have been reconnected (or it may have come loose if not correctly locked together). In that case you system may be arming; but, you are just lacking indication.
2) There are at least 4 fuses that can affect the operation of the security system. Check them to make sure one or more has not opened (best to get a copy of the electrical section of the service manual to find the list and locations of the fuses).
3) If any of the entry contacts (doors , hood, trunk... ) for the security system are not giving closed status the security system will not arm. Are any of the lights on your safety check display showing open?
4) If both doors are unlocked, when you lock the drivers door do both doors lock (and same thing from passenger side)? If the opposing doors don't lock then there might be a problem with the door switches which is preventing the system from arming.
Installing 'just' a keyless entry unit on the NSX will not eliminate problems with the security system. In fact, on my 2000 if something is blocking the security system from arming you cannot lock the car using the OEM remotes. You need to lock the car using the key cylinders on the car.
An aftermarket security system can be used to replace the security control unit on the NSX. However, wiring one of these units in correctly is a non trivial task (definitely not plug and play).
There is a bunch of stuff that can prevent the security system from arming in addition to a failure of the actual module. If there is a wiring problem or a switch contact failure then that will also prevent an aftermarket security system from arming, so installing an aftermarket system may not be the 'fix'. Until you have tested all the security system inputs you should not assume that it is the SCU that has failed. If you use the PRIME search bar you should be able to find links that will allow you to download .pdf copies of the 1991 service manual. In the electrical section of the service manual you will find a section detailing the trouble shooting procedure for the security system.
Edit
Here is a link to the 1991 service manual. Download it. It will be your best friend as your car enters the 'mature phase' of life.
1991 Honda / Acura NSX Service Manual 2.0 (nsxprime.com)
I can never find it using the Prime search function; but, if I use Google it pops up right away - go figure!
