As of Updates on February 16, 2021 they do. I had a bug in the code that wasn’t using the name during save. 
Yeah, this is me not coming up with good terminology. “Edit mode” is the time between placing your final point and before clicking the Save button (well, technically, you can click Save & change the route & click Save again - it’ll update that Route).
After you place the final point, the entire route becomes changeable - you can click points, then drag them around however you like.
I’d love to include snap-to-map, and I have some code in place for this that I haven’t released yet. The main hurdle is that I need to smooth out my use of the Map Matching service that Mapbox provides (this is the service that provides the map UI as well). The version I have now snaps the route to streets on save - so there’s no interactive approve/disapprove capability, you just save the route and then view it and hope it snapped correctly. 
In my tests, it was fairly easy to get a poorly-snapped map.
It also adds to my costs based on how much usage it gets.
Some questions re: snapping to streets:
- When, in the process of building a route, would you expect it to snap to the street? … While you’re dragging the line for the next point … After dropping each point … After dropping the last point
- Would you expect it to always snap to the street or only if you click a “Snap” button? (which inherently pushes the snap action to after dropping the last point)
- Would you expect to be able to “unsnap” your route?
- Following that, would you expect that “unsnapping” a route would unsnap the whole route or just a section?