Aye, It doesHa, 3 gates. In error state door is always closed and spawners are always off - this agrees with the conditions in the OP.
Now remember every vanilla torch(inverter) and repeater is considered a gateSoooo I just came up with a dirty bastard solution that fulfills all the OP conditions to the letter using only one gate (and flipping all the levers on the panel). Vanilla redstone is kinda awesome like that.
(I said before, if you're going for the least gates challenge, no computer for you! )
The whole floating facility is smooth basalt and hardened glass. Ended up using Iron tank Gauges to run the redwire as it can't be placed on glass or it's variants. Had to use warded glass for the buttons and the transmitters because the tanks don't transport redstone signals, nor can you place curcuitry on them.