You said the car 'would not crank again'. Most people understand the word 'crank' to mean the starter motor engaging and turning the engine crankshaft. So, do you mean that the starter motor won't crank the engine or do you mean that the starter is engaging; but, the engine is failing to start running on its own? Its an important distinction because the causes of the problem are typically different.
Based upon your description of the things you have tried, I am going to guess that the motor is cranking; but, not starting. On that basis, you have hit just about all of my suggestions as likely candidates, in particular the FIC since there was a recent case on Prime where the problem was a failed FIC on a comptech installation. If you have by-passed the FIC and nothing is happening, then you have addressed that issue.
You said the fuel pump is working. Did you do this by testing the fuel pump or did you hear the fuel pump go through its 2 second prime cycle when you first turned the ignition key to the run position (not start). If you confirmed this by listening for the fuel pump prime, that is good because it confirms that the fuel pump is running and the ECU is getting powered up and is at least partially alive (the part that controls the pump). If you did a separate fuel pump test, do the 'turn to start' test and listen for the fuel pump to prime. You will probably need to open the driver door and listen down by the driver side rear wheel. Unless your pump is getting old, an OEM pump can be hard to hear particularly with a full tank of gas. If the pump does not go through the 2 second prime, check to see if the MIL light comes on briefly when you first turn the key to the run position. The MIL light coming on is part of the ECUs self check procedure. If the MIL light does not come on, the ECU is not powering up or there is a problem with the ECU - that would be rare.
If the ECU seems to be working, the next thing I would do is to check for stored error codes in the ECU. Your car is '95 so it should be OBDII so you can use a code scanner or you can use the service check connector and count the flashes to determine the error code. A failed crank / cam position sensor would certainly leave you dead in the water with no ignition and no injector pulses. A failed crank / cam position sensor should generate the appropriate error code. NSX crank / cam positioners always seem to look like they are ready to die because they ooze gooey potting compound; but, they don't die from that. They seem to be very reliable. What does kill them is somebody pinching the wires from the sensor after some maintenance has been done. Again, that should show up as an error code.
If you have no spark when cranking the engine, check to make sure that you are getting 12 v at each coil connector when the ignition is switched to run. No voltage could indicate a blown fuse for the coils or a wiring problem. You will need to get a service manual to find the appropriate terminal in each coil connector to test for voltage.
Do all your testing with the FIC disconnected just so that you can eliminate it as a possible cause.
You talked about the car running extremely rich. Normally that would trigger error codes - mixture out of range. Do any of those codes show up on a scanner or when you use the service check connector? One cause of running really rich is fuel pressure that is too high. If the fuel pressure is really high, you can be dumping too much fuel in during the start cycle which would make starting difficult. If this were happening you would typically smell the fuel in the exhaust system. If this is the case, then I suggest a static fuel pressure test (has to be static since the engine isn't operational) to see if the fuel pressure is even in the right range. If the fuel pressure is really out of spec I suggest doing an oil change because you may have seriously contaminated the oil with fuel in addition to washing all the oil off the cylinder walls. Seven years ago at 65,000 miles the spark plugs on my car came out pretty much sparkly clean. Any fuel mixture issue that so contaminates the spark plugs on an NSX that you are having to change them regularly to get the car to start is a sign of a serious underlying issue - or a really bad set-up on the FIC. That said, high fuel pressure would not cause an ignition failure so you may have multiple factors contributing to your problems.