That's also not the whole story. The House of Commons was completely divided on the issue to the point where the Remainers couldn't find a common position and a

That's also not the whole story. The House of Commons was completely divided on the issue to the point where the Remainers couldn't find a common position and any proposal going roughly in their direction was shot down by the other parties in their own camp because it somehow didn't represent their exact idea of staying close to the EU. It was so bad that the House had long phases where literally everything that might have meant the slightest bit of progress was consistently and summarily voted down. If this divide hadn't happened, Leave would have had a much harder time. They did play very dirty procedural tricks on the House of Commons as it was to try and force things their way.