No parent can be made to have contact they don't want to have, which is hard for lots of kids on divorce. I know mothers heartbroken by men who have left and won't have anything to do with the kids, sometimes out of the blue because they've met someone else. The law is there to help parents who do want contact, not to force those who don't.
At 16, they're a minor, but not a child. The child has the right to contact if that is safe and if the parent wants it, but nobody has the right to force unwanted contact on anyone else.
The Children Act principle was to get away from the idea of "custody" where a child was controlled by an adult, and adults had the right to visitation, and over to the right of the child to maintain relationships with both parents, so the language was changed to "residence" and "contact". So the right to a meaningful relationship was to stress that it is about the best interests of the child - that the right isn't of an adult to see the child, but the child to see that adult.
That's a world away from what is being described here.
A sixteen year old has no right to access anyone else's home. A right to financial support till 18, and end of education (and possibly university support, if parents are split, under Children Act, too) yes. A right to spend time with the parent, when the parent doesn't want that, no.
Whoever is pushing them to arrive on the doorstep is not thinking of the best interests of that young person. If they're turned away it will be a horrible, and from the sounds of it additional, rejection. That's going to help nothing and nobody.