Both the loaders on the platform AND the lead operator in the upstairs control room all need to hit the button at the same time to dispatch. For the train to be sent from unload the operator hits their button and then the main operator hits a button to bring it into load.
Once the unload station is clear, the launch cycle immediately beings if a train is already parked on the launch section (Fins drop etc) the audio is triggered as the train approaches the launch. My knowledge is from 10 years ago so it is possible that due to age they are holding the trains longer between dispatches but back then that was the reason for it.
I'd also say a fair bit could be due to staff motivation. I remember one day I had a crew of entirely Parkz members working on the load dock with me and holy crap we were flying through it (We had a competition going with Arkham for a fright nights event). We only had the 1 train on the track due to tech issues but we somehow managed 8 dispatches every 30 mins (Arkham just beat us with 9).
I miss those days TBH. a good crew, and good supervisors who actually motivate their staff make all the difference.