Also, if manure steeps in aerated water, the oxygen encourages beneficial bacteria (and maybe other microbes) to multiply. Then spraying them on soil and leaves builds up the population of desirable microbes over that of harmful microbes.
Like re-inoculating soil and plant surfaces with desirable microbes at the same time that you feed them.
I was just thinking that if one had an irrigation system on a timer, you could snap the cap off a jet sprayer so that the water all came out in one tiny, fast jet. Aim that down into the big tub so that the jet shoots bubbles down through the water and keeps it aerated.
Alternatively, keep a gallon jug or scoop made from a laundry detergent jug near the tub, and intend to scoop and stir it a little each time you pass. (Then forget to!)
It's nice that the degree of aeration of any batch of compost tea is self-indicating: if it really stinks BAD like an over-used outhouse, it is probably somewhat anaerobic. That can be actively bad for root hairs since alcohols and acids build up. If it smells more like strong compost, it's probably plenty aerated.