Microbes for soil are a good thing, no matter WHAT method is used to propagate them. I have brewed, but did not find it an efficient activity.
It has its drawbacks, like everything else in life. Each to his/her own.
For folks who build a new compost pile every few months, I think a better alternative for brewing aerated tea is to simply collect freshly-leached tea from watering a new pile (I use the trench-recycle technique), and especially during a 1st turn (after initial thermophyllic heat), and also at 2nd turn to maximize collection of mesophyllic microbe populations - as soil drenches and for inocculation of new piles. Tea from all turns is high in nutritional value for plants - in addition to microbe populations.
If a composter does not properly turn (screen) and water a pile (at least monthly) an anaerobic build-up of CO2 in the pile will kill all the microbes (microbes 'breath' oxygen and 'exhale' carbon dioxide too) - and a too-dry pile will also suffer a cessation of pile decomposition since microbes need water to live too.
In my opinion, everybody who composts should collect the fresh tea from watering piles, instead of wasting it as runoff...organic fertilizer abuse.
A properly-watered 3' x 3' x 3' pile will yield about 25 gallons of concentrated tea (if a recycle technique is used) that will equal up to 100 gallons of tea that was brewed for 3 days. In only an hour or three (depending on pile size and method of turning).
I save extra tea collected in a 1,000 gallon tank and give excess away to local Master Gardener friends as a concentrated nutrition-only drench (although such tea still contains high counts of beneficial live microbes ready to propagate in their soil).
Stored non-aerated tea is not as beneficial (as fresh) tea as a pile innoculant, although most fungi/bacteria spores are not harmed by non-aeration - they simply won't multiply profusely in a non-aerated environment, and over time (just like us), organism life-cycles end - and so unless they reproduce, populations will certainly decline over time, although spores will survive without oxygen. Fungi spores (such as micorrhizal) do not even activate until contacted by a root hair, and so do not need any aeration at all - with a viable shelf life (in liquid tea) of 2 years in a sealed container. I inocculate both tea and near-finished compost with 4 micorrhizal strains.
I build fresh piles year-round, and although it rarely freezes near Corpus Christi, Texas, microscopic inspection of freshly-collected tea even in near-freezing temps has shown little decline in live microbe populations harvested from fresh (still-warm) piles via very cold water leachea - although reproductive activity in the cold (still aerated) water has declined due to the low water temperature.
I use the finish-to-harvest (FH) method with the Frame technique, and usually maintain 6 active piles 5' x 7', so harvest about a ton of compost (at 750lb/cu.yd. dry weight) monthly (1 turn per pile per month - with pitchfork and shovel), so I'm never out of a supply of fresh oxygen-rich tea, with abundant beneficial fungi/bacteria (same live microbial populations as are in the pile).
In my opinion, aerated brewing (a recent methodology) is only necessary for folks who don't compost...Or who just like to brew.
An inexpensive aquarium water heater will keep tea brew at optimum temperature and allow brewing year-round, even in an un-insulated garage or shed.
Brewing methodology does not provide any nutritional value for plants.
And when acquiring compost to start a new 'brew', ensure that it is fresh (still moist), and from near the center of the pile, and comes from an 'alive' pile (not a 'dead' pile) - turned before microbe populations were eradicated by anerobic CO2 buildup, or a too-dry pile. 1st-turn material is high in thermophyllic populations (the critters that cause a pile to heat up, and which decompose new organic material fastest) - but are almost gone by the 2nd turn, mostly replaced by a completely different 'set' of organisms. Compost from pile external areas are much less active, and may not contain a full 'set' of microbes - which would mean that the 'brew' would be population-deficient too. If the right microbes are not present in the 'starter' material, they won't be present in the brew, either - no matter how long it is brewed. Only way to actually tell, is with a TRAINED eye looking at samples under a microscope. Most folks have no idea what microbes they are propagating. Good or bad.
And BTW, before air stones were applied to microbial reproduction, the common method utilized a small water pump (the kind used with a fish tank filter) with the output hose nozzled to produce a forceful water spray onto the surface (which added sufficient oxygen) and did a much better job of circulating the liquid too.
Old-fashioned tea collection has been used very successfully for many thousands of years. Long before electricity was invented.