From my understanding, the Trots were always positive toward the EU as they saw it as a means of bringing forth international socialism. The Bennites were far more demanding of democracy, and saw the EU as a means of short-changing national sovereignty.
I think Corbyn, and much of the "new left" believe in a reformed EU. That should have been the remain argument, but instead we were told to scare people into submission - which never works.
I think the position you attribute has more in common with the Eurocommunist position mate. Stalinism remained stronger on the continent than here (where Trotskyism held more sway). The break that came led to some interesting conclusions being drawn and broad attempts to eek out spaces within the state that could be co-opted.
The traditional idea is right, that the EU exists as a bit members money club. The argument is how far does the essence of the project initially still dominate over some ongoing attempts to shift it's focus, both by the left but also by centrists (who want it to fulfil different purposes).
The difficulty with the left Remain argument is I am not really sure how you reform such an organisation. The Varafoukis position is put camera's so people can see where the important decisions are made.
It's extremely naive for me because they will just move where the important decisions are made if you are compelled to have to put a camera.
I think the left debate was very much about short term needs to long term beliefs. Paul Mason captured this quite well, by saying we should support a remain vote now for a leave vote in the future. I thought that was a useful contribution. The left have difficulty because the right have dominated the debate on leaving the EU and we are always playing catch up.
There needs to be both an attempt to combat the right but also engage in a longer debate with people who have been won over to that position. The difficulty we have is that they often require opposing positions to be taken. Outside of Paul Mason, very few are intellectually engaging with that challenge.
Corbyn had oddly sort of stumbled upon a position that while incoherent is not a bad attempt at bridging these tensions. I don't think this is through any act of genius on his part, it is purely accidental. He is basically an indecisive bloke, who can't make a decision but also have so respect for the tradition of centralism. He's a consensus man. Rather than taking a clear decision one way or the other, he sort of floats by, trying to be fair to both sides of the debate. I wouldn't be surprised at all if he was actually unsure on the EU full stop. But oddly his position is strangely nuanced, which is a great irony.
Overall I think the left should probably have campaigned for Brexit on a left populist basis. It is fraught with danger though.