Verhofstadt wan't the President of the Parliament, but he was a leader of the Liberal grouping and also chair of the Parliament's Brexit committee so I can see where the misunderstanding arose. He's a really sensible individual and, like many European policitians from the centre, is by all accounts a huge Anglophile. And, having managed to be Prime Minister of Belgium for almost a decade (not easy job), he knows how to get things done...
That said, the UK is miles off rejoining the EU and certainly from joining Schengen. The UK has always had its own frontier controls and joining Schengen would mean they were only a strong as the weakest link of the Schengen frontier.... The Home Office would always object to relying on that eg. from a security perspective. What might perhaps be possible is some move towards freer movement eg. stays of 180 days permitted rather than 90 as now???