The light wont always come on when an alternator fails internally. It may only if current is created in one direction by the alternator not running via a specific difference in battery vs alternator voltage. And the light isnt burnt out or the circuit isnt damaged etc.
Likewise i didn't say the alternator won't work if you pull the dash in all cases, although that is true for RX-7s because both the bulb and the parallel resistor for the excite wire are in the dash, so i assumed it was true for miatas. My point was simply some people have alternator issues and/or try to modify the alternator wiring and end up blowing alternators' regulators if they wire the charge light without any resistance and a diode....then wonder why their alternators constantly blow.
If dave is saying that you don't need the dash there must be some other inherent excite path on the miatas else the alternator would not turn on. If that circuit is gone or damaged and the dash is also damaged it would not turn the alternator on. So perhaps just rewiring the excite wire with a incandescent bulb and diode or an LED with a proper resistor, bypassing damaged excite circuit(s) would resolve your issue.
I work on RX-7s more than miatas but most all 3-wire alternators work the same. This is info i posted years ago on the rx-7 forums after having trouble rewiring my rx-7 (FULL REWIRE).
"B terminal is the obvious part. That's the charge wire. It...um...charges the battery...min 4 gauge IMO.
S terminal needs direct battery signal, can't go through a relay or any switch. It can go through a fuse or breaker. This is the "sense" wire. It tells the internal alternator voltage regulator what the battery voltage is and thus how much "charge" to put into the system. The closer you connect this to the battery the better your alternator will work. If you just jumper it to the charge post it really wont fluctuate voltage output to adapt to battery state and load conditions.
The L terminal which is where the excite wire connects. Yes it does power the idiot light. It does this because current flows if there is a substantial difference between the system or batt voltage and the alternator charge voltage. This terminal/wire must be connected in some systems because it is in effect what "turns on" the alternator. BUT and this is a big but. YOU CANNOT JUST CONNECT IT TO A HOT WIRE OR THE BATTERY. It must have resistance or you will destroy the alternator voltage regulator and the alternator will need to be rebuilt because it will not regulate voltage, the voltage will just rise with RPM up to 16-17 volts. I don't have to tell you that is bad. SO, the factory dash has two parallel sources of internal resistance. One is a light, the other is a standard resistor. You CAN NOT skip the resistor because if you just use a light and the bulb burns out, guess what, your alternator shuts off. So, you have to wire this terminal back to +12v (again no relays upstream for this one too) but you must also put a roughly 120-150 ohm resistor on this wire. Up to you if you want a charge light in parallel. It appears in some cases (3rd gen rx-7 and other cars) the factory dashes just use multiple idiot lights as the redundant sources of resistance for this. So if one bulb burns out the alternator does not just quit working. If you do put a charge light in parallel you need to put a diode in also or the light will be on all the time. The diode should only allow current flow TO the alternator. It needs to BLOCK current flow back from the L terminal.
I think in your case your alternator may be damaged or isnt charging because it isnt turned on. But your assumption that the alternator "stays on" when the engine is not running is incorrect. This isn't actually possible. Without the little wheel spinning on the front of the alt it's basically a paper weight, it cant be on without being driven. But, if you damaged the voltage regulator it may create a bleed. If i recall correctly when i blew one trying different things to get it to work properly (read just put unresisted +12v to the excite terminal) it self excited after that and would not regulate voltage at all."
Thus it could also be in the case of miatas that the alternators will self excite? I dont know as i have not fully rewired a miata.