`$()` creates a new quoting context, so you don't need to escape quotes inside it. That's one of the reasons why `$()` is preferred to ````, in which mixing nesting and quotes would lead to a nightmare.
For comparison:
$ echo "screenshot-$(date +\"%y%m%d_%T\").png"
screenshot-"200701_14:56:19".png
$ echo "screenshot-`date +\"%y%m%d_%T\"`.png"
screenshot-200701_14:56:27.png
You can leave the backslashes out and just use `"screenshot-$(date +"%y%m%d_%T").png"`. Or, since `%` and the others aren't special, just leave the inner quotes out altogether:
$ echo "screenshot-$(date +%y%m%d_%T).png"
screenshot-200701_14:57:23.png