There are good reasons to invest in a small aquarium pump and diffuser stone and it will set you back less than $10.
The continual flow of air through the
water does a couple of things: (1) it purges the chlorine out of the
city tap water in a very short time. You could do that the lazy way by letting the water sit overnight before using it, but that just adds another waiting step. (2) it raises the oxygen content of the water to a saturated level, so you need never worry about anaerobes hanging around. Although there are aerobic bacteria that are pathogens, it's the anaerobes that are particularly nasty (like
Clostridium). (3) It purges
CO2 from the solution, taking away a waste product that if it hangs around, can put a damper on the kind of bacterial growth you want in your compost tea.
Another way to keep compost tea from foaming is to add
biochar to the bubbling batch. Foam is indicative of proteins in the water, and biochar will absorb those proteins -- to the benefit of any microorganisms that have set up house in the biochar. I keep a 5 gallon
bucket of compost tea bubbling on the back deck and I never run out of uses for it. When the weather is sunny, I usually use it up and make a new batch on a daily basis. When we get a lot of rain, that's when it can just sit bubbling away for 3 or 4 days.