Get ready for this one Mark, it's a SABRE type of answer
In theory all public rights of way that aren't vehicular roads should appear on definitive maps of rights of way. In theory all publically maintainable highways including nearly all rights of way should appear on a statutory "list of streets maintainable at public expense". The definitive map records the level of public rights on a way but the list of streets generally does not because it is a record of maintenance responsibility not public rights. Even so the route would have to be a public route in order to be publically maintainable; the council does not have a duty to repair people's driveways.
ORPAs (other routes with public access) are a way of showing on the map routes which are included on the list of streets but not the definitive map. In other words the highway authority believes them to be highways but is unlikely to be sure what type. All they will say if you ask is that they believe them to be at least a right of way on foot but may have higher rights. In practice many were thought to be unsurfaced public roads at the time of the rights of way surveys back in the 50s. The guidelines back then were very vague as to whether these roads should be included on the definitive map and often they weren't. This has left us with the worst of both worlds where we have ways that are marked on maps but not on the ground, are frequently obstructed, no-one knows their status and if one needs maintenance the council's rights of way department will shrug and refer you to highways department who will, frankly, ignore you (I speak from experience).
So yes, the red dots do show a route which should be open to the public at least on foot. Either the council is wrong or the landowner is either wrong or lying. It sounds silly but many landowners don't know the extent of public rights across their land. They assume that they would have been told when they bought the house or mistakenly think the public can't use a road because they own it or have to maintain it. Also, council's highway records often contain errors so they can't always be said to be correct. The only way to find out for certain would be to do lots of tiresome research and then seek to add the route to the definitive map if you found evidence it was public. That is a long and drawn out process. Frankly, the whole thing is a mess.
I have to say that you have been quite unlucky, I routinely use ORPAs and have only come across a few which have been blocked off. Unless you are in a hurry those routes can be interesting. If you are only riding for leisure I guess it's not too much of a nuisance to come across a blocked path and have to turn back every now and then.