It would indeed be nice to be able to drag a protocol directly into an experiment. This is something that we will have in a calendar view in Findings 2.1 (see also: New features in Findings 2, compared to Findings 1).
It’s not something that we plan to do the way you describe, as that would really change the interface paradigm of having the sidebar for groups of documents, and the documents listed in the middle pane. However, with an experiment opened in a separate window, this could be possible to support (because you can then drag a protocol from the middle pane into the experiment in the other window). I will add that to our todo list, though I am really not sure when it would be done.
One more tidbit: when you have a protocol opened, you can insert it into an experiment by clicking on the ‘•••’ button at the right, above the protocol title, and then it will let you pick the experiment and date to insert it. Maybe that could be useful…