Gotta ask, what's your indoor humidity? That might play a larger factor than you realize, but that's a guess on my part. My rain soaked/ snow covered wood I bring in sit about 8' from the stove, and get plenty of ventilation. Plus, my indoor humidity runs around 20-24% during the winter. My tubs of chunks and uglies dry out pretty quick, usually within a day or so. I can't recall ever having seen water at the bottom of a tub, but my RH is so low, I'd be surprised if I actually saw a puddle.
Right now with the stove running, it's 42%....this will drop by the end of November...either way...wood indoors does better given that it's always warm and the air inside with a stove burning is lower than outside. I won't have any puddles given that it was wood taken from the top of the pile and the rain stopped this morning. Snow on the other hand is a different story. It tends to puddle for a period after it's brought on the wood indoors. During snowpocalypse, the wood had ice on it when brought in. I hate to think about January / February of this year for that reason alone.
I hear yah about the snow-on-wood thing when it get's really cold. I smack the splits together before loading in the wheelbarrow, echoes around the canyon like I'm beating baseball bats against one another.
I like this idea a lot. I don't have much use for a bin on wheels given my set up, but I do use an apple bin for my indoor wood storage. It's 4ftx4ftx2.5ft tall, and I stack my wood in it instead of tossing it in. Stacked neatly and above the lip, I can get between 1/4 and 1/3 of a cord in it, which gives me 10-20 days of fuel depending on the outdoor temps. Major upgrade over last year, and it really helps with the mess. You get a little debris reloading it, but realistically, I should sweep every 2 weeks anyway.
I'd lay down some 2x lumber and put the plywood on top of that. Maybe cut it a little smaller, or drill holes in it to allow water to drain to the bottom of the tub. If there's snow on the wood and it melts and the bottom is waterproof you're gonna have your bottom splits sitting in a puddle soaking up water