This is not a public right of way so it's very unlikely to be cycle friendly. It's why I have always said he walked.
Whether something is a public right of way or not does not affect its suitability to cycle. Bridleways and byways can be easy or hard to cycle on. Private roads can be easy or hard to cycle on. All depends on the maintenance in the years prior to cycling on that particular road/path or whatever.
Often footpaths can be better for a cyclist because bridleways can be churned up by horses, and byways can sometimes be totally destroyed by cars. But they are still always negotiable by bike.
Obviously it is illegal to cycle on a footpath, but that wouldn't have mattered for Bamber.
But to travel 3-4 miles from A to B on off-road terrain is easy. Even if you have to push the bike part of the way. And that would only be for short distances. And that would be rare.
Negotiating styles on footpaths with a bike is simple and relatively easy.
There are many ways to negotiate a set of styles on a bike, including throwing the bike over the styles (yes, I have done it).
Usually what I do is to lift the bike up over my side of the styles and rest the bike on the lower part on the other side, and just let the bike fall.
I've just looked at the map, and there's pretty much a footpath that goes in a straight line from Goldhanger to Pages Lane. There's some risk going through a couple of farms, but then Bamber is a risk taker, with no real sense of the consequences of his actions. There is a clump of forest nearby, and he could just walk the bike though that.
And then there is the sea wall way, which also looks easy.
I've never been bothered about how Bamber got to and from WHF, because I know there would be many ways and routes long and short between the two places, and Bamber could choose whatever he wanted to, being someone who had lived there all his life.
But looking at the route between the two places, it couldn't be easier. The two most obvious routes are both very simple to negotiate.
There are many more routes he could have taken, and there is a footpath that goes into WHF itself. There is a convoluted route back to Goldhanger, which he could have taken. But if it were me, I'd have taken the direct route and walked around the farms, or taken the sea wall route.