What advantage do you think this has over composting on the ground or in a garbage can, and then mixing the finished product with water, just before it is delivered to the plants?
Compost Tea was
recently popularized by Elaine Ingham. I'd describe her as lady who loves looking at compost through a microscope. The purpose of making tea is to greatly increase the number of beneficial organisms. We know that beneficial organisms love aerobic conditions, and need food — so we take some compost, add it to a bunch of water, force air through it, and add a little bit of food. This bubbling action increases the number of beneficial organisms by orders of magnitude compared to just making a slurry with water (your second question).
But from my research into compost tea, your first question is one that has yet to be definitively answered by anything peer-reviewed, but widely used here in California in organic farms for high value crops (think Humboldt county…). It's been difficult to measure the effectiveness of compost tea over on-ground composting because compost quality varies so widely. It's very likely that just by watering compost on-ground, the same explosion of beneficial organisms happens inside the soil since it also provides aeration and food — it's just unreasonable to count the organisms in the soil via microscope (since they are spread unevenly), while it's easy to do with a bottle of liquid (where they're more homogenous). One reason people usually prefer tea over on-ground composting is that it's easier to scale. I know a few people with 10,000 gallon aerators hooked into irrigation. That way, you get a lot of the same ease of use of chemical fertilizers without any of the chemicals.