I understand that if you want to edit the menu of unowned restuarant lots you have to buy the lot, go to the business, edit the menu, then sell the lot and the menu should stay the same. I do those exact steps and yet
every single food item is still listed on the menu as if I didn't edit it at all.

My exact steps are: choose a household -> when in the household use the phone to get sim to buy the lot -> immediately travel to lot -> click on podium to edit menu -> immediately return home after editing menu -> sell the lot as soon as the sim gets home -> travel back to community lot to test.

I find that when I go back to the lot the edited menu isn't there and every single food item is listed when trying to order food. What is going wrong? Do I have to wait a certain amount of time before selling the lot or am I doing things completely wrong? Any help would be appreciated.

You can't sell the lot, that resets it back to an unowned community NPC run lot. Simply don't sell it. You don't have to play the owner, you could even have them live off to some far corner of the lot and never play them.