Sometimes I pour the mix back and forth between two large buckets or mix it up in a wheelbarrow like stir-frying, but with a small shovel. And I always moisten it before pouring into cells or pots.
For years, I used far too much water to be SURE it was all moistened, and trusted that any excess would run out. But over-watering a peaty mix tends to rot seeds, impede roots and cause damping off.
And when there's very much peat in the mix, water DOESN'T run out on its own! I see what you're saying about squeezing and squishing the mix to push water into it and then wring excess water out of it, but I want to "fluff up" my mix, not compact it.
So went another route and started adding lots of grit and Perlite. Lots. But that made it even more expensive.
Now I use a lot of screened bark and just a little professional peaty mix.
That way, I achieved my goal of having any excess water RUN OUT THE BOTTOM of even small cells or tiny pots. My mix DRAINS.
Maybe I have to water often, or keep a bottom-water-mat damp, but I no longer lose whole trays of seeds to root rot or damping of!
