What's interesting about the 'SCAG is better than Extra Attack' orthodoxy from the front page is that:
A.) It assumes absolutely no magical items. The core assumptions of the book assumes that, but the hardcovers, softcovers, and canned oneshots sure as hell don't assume that.
B.) Even without the assumption of Magical items, the Fighter / Abjurer ends up falling behind assuming the arrangement of SCAG + Haste Attack + Off-Hand Attack at higher levels when you throw in Song of Victory. The Fighter/Abjurer can sort of keep up with GMW, but then they lose out on AC and they trade a good primary stat for a mediocre one.
C.) At VERY high levels, Bladesong can be used with Shapechange. But lol level 17.
But I think the focus on melee DPR is wrong when it comes to the Bladesinger. See the next quote.
Nathan Mitchell said:
A tank doesn't just need to be able to survive taking hits, a tank has to give his enemies a reason to hit him instead of his allies. A good tank has some way of punishing enemies who want to attack different targets.
If a Bladesinger really wants to draw hate, the best way to do that is to drop a concentration spell that wrecks the battlefield. Once you bust out the Polymorph or Fly or Evard's Black Tentacles or Wall of Force or Banishment, trust me -- the enemies are going to start gunning for them. This is a huge problem for other wizards, especially once nasty attacks start becoming standard for ranged enemies. I played through Storm King's Thunder, and I frequently saw the party sorcerer drop concentration on a Wall of Fire or Telekinesis or Sleet Storm due to getting a boulder or greatsword to the face.
This is why I think that the suggestions of the Bladesinger to draw hate by focusing on melee DPR are highly misguided. If you want to draw hate, just do what wizards do normally.