If you click on “edit” on the main openstreetmap.org website, go to the "Background Settings" (Shortcut:B) on the right side of the editor and scroll down you will see a checkbox for “TIGER Roads 2023”, click on that, you will see the respective road names on there, while the alignments from there tend to not be the best the names are. Also if you go to the following site (https://gis-mdc.opendata.arcgis.com/datasets/b0107c2e893a4ff78367837cf80bdfaa_0/explore?location=25.725668%2C-80.252825%2C16.55) you will also see similar road names in the area. TIGER is from the census bureau, they are as official as it comes. There are other databases on in the Miami dade open data hub that also show the names similarly. Those are the official names, I have gotten them from sources that have all the resources they need to be accureate. 

The roads here are way to wide to be service roads and play way to an important role in role in the road network (especially Brooker street) to be service so that’s just plain wrong. Brooker and Charles Avenue in fact do pop up on the mimi dade oped data hub. 

They are not private as these aren’t driveways or part of a gated community. There isn’t nothing physically stopping someone from driving through. Brooker street especially as it has non private segments further north whose traffic flows directly into the segment you keep marking as private without the road getting thinner or a gate being put in the way (this simple logic also applies to the fact that it is residential). I can half accept Charles Allen Avenue and Charles Avenue being private as they do seem to have some signage indicating something of the sort on bing streetside but Brooker Street and William Avenue have no such signage on the bing street side with the Brooker Street also having the role it does in the network as described above.

as for the "City of Miami municipal road atlas", I counld find a single link to buy it so I assume its either out of print or hard to get and as such is a source that couldnt be checked for validity and licence complience with OSM and thus I doubt its usefulness as a source for OSM contrinution. The data from the property apraisers office of miami dade sais nothing of the road network in the area, data from the apraisers office would certainly be useful for mapping addresses, buildings and landuse in the area if its licence complient with OSM but it shows and sais nothing of the road network in the area and thus doesn't really help in this senerio. 

As an example of the gate on either side principle you only have to look slightly north as vertexes https://www.openstreetmap.org/node/7509485821 and https://www.openstreetmap.org/node/7509485798 enclose a set of roads that are all private, if those werent there how private would it be. As for the "St. Alban's Day Nursery" it is clear that the parking on the side of the road is private from the bing streetside and the property apaisers office but that doesn not aplly to the road in general, drivers backing out of the parking may have priority but that doesn't make it private. Just because there are private land uses next to a road deosnt mean it is access=private, it can be used by cars passsing through an thus isn't private. 