Wouldn't the ammonia and nitrite (I think that's what you're referring to) just get converted to Nitrate which would get removed by water changes with the vaccum?
Axolotls poop quite a bit. It would take a long time for the poop to vanish into nitrogen compounds... ammonia -> nitrites -> nitrates -> down the drain with a water change. The water changes vacuum up a lot of the poop but it is hard to get everything... especially after it has started to break down after a week of sitting.
Keeping the little critters is a bit different than keeping the tropical fish I am used to. They are more sensitive to water quality than our cichlids and guppies. They also require a cold tank ~55f - ~64f. Removing the waste between weekly water changes will provide a healthier environment than letting it sit. My daughter pretty much has to do water changes herself when Doja is at her mothers.
When Nicole originally asked for an Axolotl a couple years ago I batted down the idea because she knew nothing about what it really took to keep an Axolotl.... by herself. Nicole developed a folder full of Axolotl care information. She learned the background of Axolotls and about their natural habitat. Nicole's 11yo German Sheppard at her mothers house had to be put down due to age which prompted me to make a deal with Nicole's mother to help fund the Axolotl tank (Nicole had to make money and contribute towards a portion of the cost). After 3 months of cycling the used 55g we found a couple larger Axolotls on Craigslist that were not being cared for very well. One of the Axolotl's turned out to be a female and laid eggs which Nicole decided she was not equipped to care for so we took the female to our trusted LFS.
Nicole performs the water changes by herself at her mothers house after a bit of startup help from me. She also has a very good understanding of the nitrogen cycle. She is very adept at doing water tests and making decisions about the condition of the water in the tank. She has overcome her fear of cutting up live nightcrawlers to feed Doja. So far it has been an excellent learning experience for Nicole and has taught her a good bit of responsibility.