I handle it very differently.
First, I never put a puppy in a crate in the same room as people are in. Every time you roll over, or get up for 'reasons' in the middle of the night you disturb the pup. Then he cries, you think he needs to go out, so you bundle up and out you go. Then, back in the crate, you hope he settles, and all can get some sleep.
This happens several nights in a row, and you are now training the puppy to go out in the middle of the night. A vicious circle.
Assuming the crate is the correct size for a wee one 10 weeks old, and you put him down after a spin in the grass, last thing, he should be fine.
I never worry about an occasional wet crate for the first several weeks. I use an old towel for bedding, and if it is wet, oh well. It is kind of like a diaper at that age. He doesn't want to wet, but sometimes mistakes happen. Ignore it, wash the inside with cleaner, put in a fresh towel and move on.