True for motorways, but even on busy dual carriageways sometimes the planners just say "the footpath crosses the dual carriageway on the flat", ignoring that this comes pretty near to being an effective closure of the right of way; there are some on the A14 near me like that.
There are a couple of instances of this on the A45 just after you leave Coventry going towards Birmingham. Some of these were later closed off entirely (the A45 has been there a long time, but some roads I used to cycle on 15 years ago and now closed). There's one or two that I would take my chance on when cycling, but there's simply no way to cross safely as a pedestrian because of the curvature of the road restricting visibility and the speed limit. One nice change is that the new road junctions to Birmingham airport now mean that there are more cycleable bridges again.
As to the GP's comment, since I got a gravel bike, I've been exploring a lot of random routes that are basically underused footpaths (because they don't really go anywhere useful) that cross the motorway in delightfully over-engineered ways, for instance a narrow path through a field of crops, a stile then a nice tarmac'd two lane bridge and a stile to another field on the other side.
This forum thread lists some other examples: https://www.sabre-roads.org.uk/forum/viewtopic.php?t=41259