The Spin Doctor rescue minipumps are similar design as the Lezyne Pressure drive minipumps: they both have a flexible hose that terminates in a screw on chuck. This design is nice because you only have to hold the pump itself and not hold the pump on the valve where you risk tearing the tube at the valve stem base.

I'm assuming your tubes use presta valves. The way a presta valve works is that the pressure inside the tube is higher than atmospheric pressure and it's only this pressure differential that seals the valve shut. When you are pumping air in, what's happening is that you are increasing the pressure in the hose from the pump to the tube, so the higher pressure in the hose causes the valve to open and air to enter the tube until the pressure on both sides of the valve are equalized. And when you pump the piston again you increase the hose pressure and put more air in again. As soon as you start to unscrew the air chuck, the open-to-the-air side of the valve should rapidly drop to the outside air pressure (that's the "escaping air" hissing sound that people hear when they unscrew the chuck), but the high pressure of the air inside of the tube causes a pressure differential that shuts the valve closed without much air escaping from the tube at all.

It may be possible that when some people unscrew the chuck they aren't pulling it straight off the valve stem, and if the chuck is somehow pushing in the little presta stem or the nut attached to it, they may be pushing the valve open and letting air from the tube escape. But this falls in the realm of "operator error".

A lot of people like to start with the minipump and top off with CO2 because many pumps can't realistically pump road tires up to the pressures many people like to ride at. I struggle to get them up to 50-60 psi as the pump gets too hot and starts to leak at one of the seals somewhere because I'm impatient and pump pretty quickly.