The way to find out if you have a choke or an enrichening valve is to take the fornt cover off the air box (might need to sdo that to adjust the idle settings anyway) & see if the choke (key push in your case, older remotes had a separate sprung toggle switch under the key) and see what happens.
If it pulls a plate across the air inlet, it's a choke in the traditional sense, and so will probably need to use the fast idle lever. if it moves something on the side of the carb, but you see nothing move in the air passage, it's an enrichening valve, which means if you use the fast idle lever, yiou are letting more air in so you might as well not have bothered choking it!
What they may have done on the demo was started on choke alone then very quickly (once running) opened the fast idle to keep it running & warm up quickly.
Although it sounds like you have a problem with your ASIG switch, which i think is in the starter circuit and is held closed by Neutral, opens when in gear. If it's not that next stop is either the keyswitch itself or the solenoid. I'm pretty certian if no deadman it would still crank, just won't fire.