Government's Parliament Majority
The EU would rather not want Brexit, so the thought of "it's pretty much controlled by the EU" is interesting.
A European law officer gave an opinion that the UK could stop the exit process by saying "I was only joking" (ref). However, I doubt that'd happen without someone being metaphorically ripped apart by protesters (or literally).
I have no idea what the EU wants. Nor the voting public. It's hard to say who does know. MPs talking in the debate recently have firmly announced that they do not agree with the outcome of the referendum their area, even when the majority was high. Whatever happens, I feel that at least 49% of the population will be unhappy about it.
It's hazardous for the representatives in a representative democracy to openly defy the will of the electorate, so some grounds is needed to make it appear one has not done so. I could maybe see a new referendum being held at some point, which would potentially allow politicians on both sides to avoid taking personal responsibility for the decision. Like impeaching Donald Trump, it's a very tricky business that, if it's to stick, would have to be gradually sidled up to, not making any sudden moves that might leave behind part of the needed coalition as one coaxes it to follow along. It can be really hard to judge where one is along that sort of curve; to some extent, reaching a point on the curve becomes a matter of successfully projecting confidence that one has reached it, so the judgement of reaching it is not separable from the act.
A new referendum might actually be best. A lot of young people didn't vote because they thought it didn't matter. There's some evidence that the majority of the British population didn't want Brexit, but not enough to override an actual vote (I wouldn't see that as justified in the face of anything short of criminal tampering with the election, hard proof that the results cannot be trusted; here, no one's doubting that the votes of people who bothered to vote were counted accurately). "Let's vote again" seems the simplest and most direct way to deal with that. If the deal falls through, they would have sufficient impetus to call for one.