Brexit was a popular vote, with neither party actually supporting it. The interest in Brexit also grew against the wishes of major parties, industries and most centres of power. A reason parliament can't manage it decently is because parliament was and is mostly against it, including May.
This is not like ip laws, where interested lobbies co-opt the parliamentary systems with as little popular involvement (or knowledge) as possible. This was/is an actual division of opinions among the people. It wasn't imposed from above.
I hope you guys stay in, or at least close but I think it needs to be done in a way that respects the other side too.
There were about 3.4 million non-UK EU citizens, many of them settled adults of voting age. They would have voted if they had migrated to the UK and become citizens. But they were told all their lives, that EU citizens don't need to become UK citizens, they have equivalent full rights in almost every respect without doing so, so they lived here in a fully settled way akin to citizenship. Sometimes for decades, with children and grandchildren in the UK. Some of them had respected positions in government.
Any of them, of voting age, could have registered to vote in the referendum if they had known sufficiently far in advance. That is, they could have applied for UK citizenship to do it - though not in the timescale in which the referendum actually occurred.
But EU citizens were discouraged from adopting UK citizenship because there was no need. This led to the perverse situation where non-EU immigrants had the vote, and EU immigrants did not, through no fault of their own.
The UK citizens, of voting age, who were not allowed to vote were all abroad. I expect most of them in the EU.
I'm pretty sure both groups would be pro-EU, and that 3 million number is more than enough to change the result.
So I would not say the Brexit referendum result of 2016 was even the popular result, for any reasonable definition of democracy in my mind.
The inability to force a clear Leave manifesto is the root of the disaster. What "should" have happened is either a Leave PM, or the 2017 election should have been run on a clear Leave manifesto specifying a realistic deal to be sought against a Remain opposition.