I put up a small proof of concept to demonstrate @jotpe's proposal on Github:
Just click on some street with a name and see all parts of the street highlighted, regardless of how fragmented the underlying OSM data is. Small gaps are also ok.
Caveat: a bit of patience may be needed, in particular during London night time. As dev instance doesn't run on fast and shiny SSDs, other heavy load jobs have quite some impact. The night window starts at about 01:00 (UTC+0) and finishes at about 08:00 (UTC+0).