Noticed that the FDOT GIS files for state roads was recently updated to show SR 777 along River Road in Sarasota County from US 41 at North Port to I-75. The attribute data lists it at 5.84 miles.

Checked the Florida Straight Line Diagrams and the entry for
SR 777 (Roadway ID 17000064) was revised on August 19, 2020. The net roadway length is 5.819 miles, which extends to the turnaround north of I-75 (where a rest area entrance was located at one point).
Drove SR 777/River Road yesterday, and the route is not signed. Nor is it acknowledged yet on guide signs along Interstate 75 southbound.
Edit - Went ahead and created a page with the photos on AARoads.Will be checking on subsequent trips to Southwest Florida and will post when I see SR 777 marked.