And to remember the key question:
Q. "does this make the BS OP?"
A. Well, not really, but it IS strong. If you want to call that OP, fine, but then do it because it overshadows Wizards with poorer defense, not because it overshadows Fighters or even comes close to."
Here's the thing about overshadowing wizards with poorer defense; they usually beat the Bladesinger on other areas, especially around level 14 or so.
Abjurer has a lot of the 'don't worry about me, lads' qualities of a Bladesinger but also comes with the ridiculously powerful feature of adding their proficiency bonus to Dispel Magic and Counterspell.
Conjurer isn't exactly OP right now, but after Xanathar's comes out and releases a bunch of new fiend-summoning spells we'll see.
Diviner needs no explation.
Evoker wizard, especially with Jeremy Crawford's interpretation of Empowered Invocation, does huge amounts of reliable damage especially in conjunction with Hex and Contingency. I think only the Sorlock Eldritch Blaster can consistently beat them without dipping too far into their stock of spells.
Illusionist's Malleable Illusions and Illusory Reality is outright broken if they use those abilities to their full potential.
Necromancer, if you abuse Animate Dead and your DM lets you switch out your skeletons' equipment, is just evil.
Transmuter... eh, no killer app.
I don't need to elaborate on the Theurgist.
I don't need to elaborate on the Lore Wizard.
Now, I'm not saying Bladesinger is underpowered or even not-OP. But at higher levels? At around level 11 or so, I often find myself wishing I went with a Diviner or an Illusionist or an Abjurer. The simple fact of the matter is that even with goodies like Purple Wyrm Poison (SKT has a place that lets you milk them for it if you're bold) and Flametongue Shortswords, spells will just outshine your melee capabilities. So oftentimes, you'll have wish you had something to enhance your spells rather than your melee capabilities.
That said, Bladesinger does shine in two ways. First of all, the wizard spells assume that the caster doesn't provide much in the way of melee interdiction. This was a big deal, especially at low level. Being able to trap a group of foes in a corridor caught in a Web with me on one side or corner and my buddies on the other side is a trick that never got old. Being able to get right up in the face of some caster jerk and force them to either Misty Step out of the way (and get stuck using a Cantrip) or eat a counterspell and/or melee retaliation is good, too. It's becoming less of a deal as I lean more on spells, but I'm hoping that this Ring of Free Action will breathe some life into the tactic.
Secondly, AC is still the best defense in the game and Song of Defense/Absorb Elements does stretch out their durability. Even with the reduced workdays of actually-played 5E D&D, you can still run dry on resources if you're not careful. Being able to drop a Greater Invis or a Haste on the melee characters while not being worried about painting a huge target on my head is relaxing and it HAS happened to me when I play other casters. You do not know pain until Stone Giants decide to gang up on the caster who has half of their team trapped behind a corridor Wall of Force.