It's correctly tagged as highway=service + service=driveway + access=private, as it was the first time you deleted it. It was then re-added without access=private tag missing, because it is visible on aerial and street side imagery (and not just that provided by Bing). It is in any case unlikely that people using OpenStreetMap-based tools would be directed into a dead-end road even without appropriate access tags.

Your access road is shown on some current Ordnance Survey maps as a normal part of Strawberry Hill Road (same width, street name labelled), which may be more likely as a cause of people believing it to be a public highway.

It is also included in OS Open USRN (Unique Street Reference Number) with the same Designated Street Name ref. 22403854 as the main part of Strawberry Hill Road.

In the OS Open Names product, the Named Road point for Strawberry Hill Road (ref. osgb4000000030496469) is located on your access road, rather than on the main part of the road as I would expect.

I suggest that you contact Ordnance Survey for an explanation of why this is the case. The highways department of LB Richmond upon Thames and your ward councillors may also be able to help.