We can broadly distinguish three different situations when adding access tags to highways which are alos also Public Rights of Way (PRoW).
**Firstly**, Normally certain OSM highway tags can be presumed to imply various access tags, roughly along this lines:
- highway=footway : foot=yes
- highway=cycleway : foot=yes, bicycle=yes
- highway=bridleway : foot=yes, bicycle=yes, horse=yes
The implied defaults may vary by individual countries but are usually fairly consistently mapped within a single jurisdiction.
In addition in England and Wales designation=public_bridleway **emphatically** implies foot=yes, bicycle=yes, horse=yes.
**Secondly**, use of the highway=path tag does require that access rights be made explicit, even for foot=yes. In general I would only tag a path with no known access rights with access=private, but for those with access rights specific each one.
**Thirdly**, use of a generic access tag is appropriate when the highway tag is not one of those typical for a PRoW (e.g., a driveway to a house, a farm access road, or an agricultural track).
It seems in this case there is a confusion: the land over which the highway runs is private. Usually the formal written designation of a public right of way will include the notional width of the relevant path. Here is an example where the PRoW as specified is narrower than its apparent extent and the landowner has requested that people stick to the path: [photo](//commons.wikimedia.org/wiki/File:Gotham_Bridleway_No.1_-_geograph.org.uk_-_739058.jpg)
Thus there are three potential situations depending on which highway tag has been used to map the way. The addition of an access=private to the first and second categories mentioned above rather complicates the tagging for no apparent additional informational value. As the OP states it has the consequence of negating any use of the standard CartoCSS for planning walks in this part of England (although recent style changes have made it less useful for this anyway).
So broadly speaking I feel this is over zealous tagging. It is not wrong, but like many such tagging schemes has the effect of making the data harder to use. Furthermore it is highly divergent from the usual way of representing such cases in Great Britain (85000 ways vs. 1300 (August 2014 data)). Normally such a major change from the existing consensus should be discussed up front on talk-gb.
This usage is not generally suitable for Public Rights of Way in England and Wales for the reasons stated above. OpenStreetMap tagging is mainly not about providing a legalistic precise view of the world, but a pragmatically usable one ('perfect is the enemy of the good enough').
**I'm concerned that that this way has been subject to an [edit war][2]. Edit wars do not reflect well on any of the parties involved. Please consider your mapping practices before making any more changes of this sort.**
[1]: https://upload.wikimedia.org/wikipedia/commons/e/e3/Gotham_Bridleway_No.1_-_geograph.org.uk_-_739058.jpg
[2]: https://wiki.openstreetmap.org/wiki/Disputes