In the search function under inaturalist.org/observations/userxyz a pop-up calendar function, which exists on other parts of iNat, would be a comfortable option. Right now one has to select Day, Month and Year separately.
Just a note that this whole page is one of the older parts of the site. I assume whenever they update it they’ll include a typical calendar style date picker.
I’m going to close this. Likely will be added when the page is revamped, but it’s not a priority now.