My limited civil and military experience was that we were never 'late' getting to a scene - there was always a long period sitting on the ground while things got organized, etc.
The only exception was when we got called out to remote areas where the ambulances couldn't get to quickly, or where there wasn't already someone taking care of immediate first aid. But these were not the majority of the calls.
Even then, I doubt that there's more than a couple of percent of patients for whom the time of arrival of the helicopter is critical for survival.
(Does anyone have any stats on this?)
What difference would a couple of minutes later make to the whole thing?
Don't rush the checks, don't rush anything - go as fast as you can when airborne, but if you're a minute or three late getting off the ground and safe, it's better than on time departure, but not prepared.
Do you see the airlines hurrying their checks when they're late? And they have a lot more money tied up in being on time than the EMS business.