Why do Nepalese authorities care about Everest climb fairness?

It is not straightforward to fly a helicopter to these altitudes, land, and take off again. It's risky, and the results depend on the weather, which can change without warning. The helicopter is operating near the limit of the thinnest air in which it can fly, and it's less maneuverable than normal. The Nepalese army used to do helicopter rescues on Everest, but now they're being handled by a private company called Fishtail Air, which lands as high as 6400 m. Landing that high is only possible in good weather. In appropriate conditions, helicopters can fly higher than the peak of Everest, but they can't land and take off that high. There have been three incidents in which Fishtail helicopters crashed while trying to rescue someone, as well as many crashes in the area before Fishtail started operating.

For a tourist to fly in to Everest base camp by choice is stupid on so many levels that it's not even funny. At these altitudes, it's all about acclimatization. Flying in on a helicopter means they lose that chance at acclimatization, which makes them more likely to have serious problems as they continue climbing from there. They're also missing out on a scenic and memorable hike, which in fact is probably the only part of climbing Everest that will not be physically miserable.

It seems like they should not care how one spends their private time and private money for whatever activity they wish.

A good analogy would be if someone wanted to do a helicopter landing on the top of the Washington Monument. I don't think the park service would be very happy about it.

The ideal is for mountaineers to be self-reliant and independent, but also that if the need arises, they do everything they can for other people. In reality, when you're at high altitude in the mountains, you may be highly interdependent with other people with whom you're sharing the mountain. If porter A gashes his forehead, and climber B from another group uses up her sanitary napkins as bandages, she may have to ask around to see if anyone else has any. This idea of helicoptering in creates more risk for the person doing it (crashing, not being properly acclimatized for later climbing), and if those risks turn out badly, it has an impact on other people. Guides, climbers, porters, and doctors are all going to be called on to deal with an emergency situation, and they will put themselves in danger, if necessary, to deal with it.