A filter would help. An air pump doesn't directly add oxygen to the water. Surface agitation adds oxygen to the water- the only reason an air pump works is because the bubbles break at the surface, causing surface agitation. Air pumps also create current in the tank. If you're going to add an air pump, I'd recommend adding a filter. You can buy a HOB or internal filter for pretty much the same price as you'd pay for a decent air pump, or you could go with the air pump, and put a sponge filter on.
I find it hard to believe that a tank with water movement or filtration and a stock of axolotls has 0 ammonia, 0 nitrites, and 0 nitrates. I find it VERY hard to believe. The fact that your axies gills are tiny supports that opinion. One of my axies had part of its gills nipped off by a tank mate a few months ago, and it grew them back in just a couple weeks. Now, that's not to say EVERY axie's gills are small because of water conditions- i have one axie who completely lost his gills, even though parameters were perfect. He spent about 4 months in a tub with 2" of water and a land area where he spent most of the time on the transition between water & land, before he finally returned to water & began regrowing gills. But more often than not tiny gills are a sign of poor water conditions, and if it's happening to a whole group of axies, it's pretty likely that it's the water conditions causing the problems. Were I you, I'd try a different test kit- yours may be bad.