Here's the thing, no matter how good every other aspect of BS is, this is an absolute deal-breaker. You have basically just said that a Bladesinger can either cast spells in combat as a full wizard, or pull a rapier and do 1/3-1/2 the damage that a melee character would normally be expected to do instead. Why would anyone choose the latter option?
It doesn't matter if AC is very good, or if movement is very good. That's like trying to sell a car with no engine by trying to convince the buyer it's worth it because of the new brakes and great sound system. Sorry, the car needs an engine or those other things don't matter.
Definitely the low damage is a fact. The PCs would like to limit the consequence of attacks back, and traditionally we've understood forcing damage to be the route to that. A typical encounter at say level 6 will have foes with about 300 HP. Any two members of a party of four probably deal out 50 damage, so if they are the only damage dealers the combat will go 6 rounds. So long as we're agnostic on matters of style, if
defenses can decline the consequence of attacks back sufficiently, the importance of one more character dealing damage becomes minimal.
Or to put it another way, damage has a diminishing return in 5e. Above a threshold, the PCs are winning anyway. So the question becomes, how efficiently did we decline any consequences in return? I accept that isn't the traditional way of looking at things, and I've had to accept some blinkered input from some posters due to that.
First thing I would mention is that I don't understand your facination with blur for BS. The BS already has a good AC, and can use mirror image and shield to become exceedingly difficult to hit without use of concentration. Remember that car with no engine? Blur is like upgrading the speakers. Engine! You need a working engine! The sound system is fine.
When we do the maths using probability density functions, we find Blur is simply far above everything else for defensive value conditioned upon how good your AC is to start with. Mirror Image is perfect if we don't anticipate too many incoming attacks per day. The low AC of the images makes it less efficient if we do, and doubly so if we need to stack it with something else which could also take two rounds out of combat (I assume an average of one for BS). Cantrips like Booming Blade and Lightning Lash tip BS toward a tanking role, not damage dealing role, in melee. Blur powerfully enables that.
Is it right that a key lense for your analysis is damage dealing? Mine is efficiency irrespective of how we get there, which is broad and time-consuming to understand and resolve. I think BS is more efficient as a tank rather than a damage dealer, and I think most parties can put out enough damage among the other members that if BS can tank successfully that's melee-mission accomplished.
Here I feel we could both be right because for me it's the flexible role of gishes that makes them most appealing. As BS I can switch to full caster and buff, debuff, AoE and CC, I can switch to reasonable damage dealer with Haste, I can switch to fantastic tank with Blur.
I'll take the 3rd level spells, unless of course I want to use a weapon in combat instead of spells, then I NEED the Paladin levels. Poking someone with a rapier and doing a fraction of the damage that a party member using a weapon should be expected to do isn't OK when you proclaim, "But I have 3rd level spells! I just don't cast them because I'm busy using this sword really poorly."
Through tanking rather than damage dealing, BS can serve the party very efficiently. So much so that it ends up with high-level casts free to do whatever is needed with. Hmm... it's almost like there is a paradox here. You're saying - BS is bad at damage. We can make BS better at damage by throwing away our 3rd level casts. But what if we reject the paradox and say BS is great at tanking. So great, that we get to keep those 3rd level casts and use them for whatever we like!
Again, the lense we choose informs the analysis. If my yardstick is - be good at melee damage or go home - then BS is a bad choice. If it's - be good at efficiently winning encounters or go home - BS is top-tier.
They succeeded in making an excellent defensive wizard, but not a fighting wizard. "Fighting" would suggest that it would not be a terrible tactical option for a straight BS to pull out their rapier and attack with it, but currently, that is the case. There are fixes, which I've discussed in my review, but they involve multiclassing.
Agreed that this is what they did. I don't think it needs fixing. Or more emphatically, I think fixing it could raise the question - why bother with other martials?! If BS could tank as well as it can now
and deal effective melee damage
and keep its wizard caster levels... !?
We could contemplate a fix that went - BS converts spell slots into melee damage. That seems fine because it means the more we care about melee damage, the less other stuff we do. But this ignores players as creative, intelligent beings. They'll pay out those slots when it is ideal to do so, and keep them for wizardry when it is better to do that. So we'd need to lock it in somehow by taking those slots away from casting in advance, rather than allowing it to be on the fly.