I run air stones in every freshwater tank because I have seen them save entire tanks when filters decide to stop working, which does happen.
If only bichirs are in the tank they have a much better change of making it without any active aeration because of their ability to use atmospheric air. That said most bichirs are in tanks with at least one other type of fish that would not be so lucky. They certainly won't hurt, but could save a lot.
The purpose of air bubbler is to agitate the water surface, increasing dissolved oxygen content in the water. When a bichir swim up to the surface to gulp for air. It is taking air directly from the atmosphere.
I run he air-pump attached to a (large) sponge filter. These are very cheap and are excellent at biological filtration. (There was a 4 month period where I was running an overstocked 90 Gal tank exclusively on two large sponge filters and weekly water-changes.)
So, while an air-stone in a bichir-only tank is not going to be vital, an sponge-filter only helps the overall nitrogen cycle (even when you have another filter).
I think every one is assuming the filters will never stop running, which is a dangerous assumption. I have seen tanks saved by the air stones that had been running when the main filtration stops for one reason or another. As stated, bichirs have a great chance if this happens, but unless there are no other fish in the tank IMO it is still worth having an air pump in case the filtration evers stops for some reason.
i use a sump with Rio Hyper Flow 20 for my filtration, but i still have an air-stone bar, just to help circulate the water and avoid dead spots.
also on some Sump setup, they have air-stones on some compartments to re-oxygenate the water, as BB processing are aerobic to remove nitrate, nitrates, etc..
The denitrifying bacteria that remove nitrate are anaerobic and only grow inside rocks (like live rock or Seachem Matrix). They can't live with the same oxygens levels as the nitrifying bacteria that consume ammonia or nitrite and require oxygen.