I was having the same problem as MudasGrid. 
I had a long post and a file ready to upload and decided to go back through my saved versions of the workbook one last time. I could have sworn I had it working earlier, and eureka!
In the protect worksheet screen in the Allow all users of this worksheet to section I did not have the edit objects box checked. Once I checked that I could use the searchable dropdown as intended with the page protected. I hope this helps someone else. I spent way longer than I am willing to admit trying to figure this one out! I think if the two of us had the problem then there are probably others trying to figure it out too, hope this helps someone!