The standard belaying method taught to beginners at a gym these days is PBUS, which you can find out about by googling. There are other methods you'll sometimes see used, usually by older climbers, but the method you were taught has never been a standard one. So to the gym employee, it just looks like you have no clue what you're doing.

The main problem with your method is that's it's extremely strenuous and awkward. It's fine when there's almost no friction in the system, but as soon as any friction is introduced, e.g., because the rope is running around a corner, it becomes incredibly hard to pull in rope if you only apply force from below the belay device. The reason for this is that a belay device is a force-amplifying machine, and it will amplify the force *in either direction*. That is, let B be the tension in the brake-hand strand and C the tension in the climber's strand. Then the rope will stick if both B/C and C/B are less than some fixed value, which is typically about 10 (Fuss and Niegl, 2010). Normally C is greater than B, but when there is a lot of friction in the system and you're trying to pull in rope, B will be greater than C by the same factor. In the PBUS method, we use the left hand to reduce C to nearly zero, so that B can be very small. In the method you were taught, if C becomes fairly big, then you're just fighting the device to produce a big enough B.

I tried your method with a rope that I happened to have set up for a toprope, and I also found some secondary problems.

In order to reach down far enough to get the bottom hand under the top hand, I had to do one of the following: (a) hunch over, (b) lift the belay device a little bit above the belay loops of my harness, or (c) keep my hands inverted above the belay device. Option a is incredibly awkward and looks like the body position of a newbie who doesn't know how to belay. Options b and c put everything in a position that is not the right position to hold a fall.

The other problem I found was that when there was enough friction in the system to make me work hard to pull in rope, pulling in rope became an extremely slow process. That meant that I was not spending most of my time in the passive, comfortable braking position, ready to catch a fall. You want to be in braking position almost all the time if possible.