Because the Bladesigner, as a melee class, comes off as underpowered when the focus is on the melee capability - primarily because it is hard to elevate damage and maintain defenses due to all of the restrictions on concentration.
From a math point of view, this was true with the SCAG bladesinger, but it is no longer true with the new version of the bladesinger extra attack. At 5th level a bladesingers attack action is landing an extra d8, at 11th level 2d8, at 14th an extra 2d8+2xInt and at 17th an extra 3d8+2xInt. That is before you consider secondary damage which is easy to land as a bladesinger if you are using Blur or PEG against an enemy that is effected by it (or for that matter Haste).
In addition to the above, when caluclating damage with blur or PEG you also have to also consider that AOOs against you do not matter because when combined with shield or silverybarbs, they will almost never land in tier 1 or tier 2. This means you can move at will and force the enemy to move or give up on attack actions. Usually (not always) you are going to land secondary cantrip damage if you are a bladesinger using extra attack with booming blade and blur up. That is another 1d8 at low levels and 2d8 in levels 5-8. If you use Shadow Blade or Hex or Spirit Shroud or something like that you will do the same or less damage, while getting hit more often.
From a straight math point of view it is not underpowered. In play the thing that really makes it underpowered in melee on the base class is the lack of weapon proficiencies and inability to use strength-based weapons if you play with point buy. This puts you in a situation where, using only the class options, you can't use most of the magic weapons you find well unless it is the one you picked with training in war and song. The same thing is true with a lot of power melee builds using Great Weapon Master or Pole Arm Master, they tie you to a very small number of weapons to make them work well.
You can fix this with race though on a Bladesinger. Playing a Drow gives you the exact weapon proficiencies you want (as does playing another class with weapon proficiencies and switching them to what the Drow gets). This makes your weapon selection as good as any Dex melee character. Likewise playing a V. human or custom and taking weapon master at 1st level to boost dexterity also gives you the weapons you want without putting you behind on the ASI tree. Finally, it is fixable if the DM lets you buy magic weapons, which allows you to just buy the exact weapons you need.
Blur reduces the damage you take. That reduction only matters in D&D if you would have gone down in combat without it.
In tier 1 and tier 2 blur or PEG combined with shield and bladesong nearly eliminates the damage you take from attacks. In tier 3 it severely reduces it.
It is not uncommon for a bladesinger to be in a position where an enemy needs a double 20 (20 with disadvantage) after you already put up a shield in order to hit a bladesinger. If you have not put up shield yet, even that roll is going to probably miss due to silvery barbs.
Blur or PEG will cut the damage you take in teir 2 by about 90% and if you are always in melee, that will matter unless you have other characters that are going to heal you or have spare actions to spend drinking healing potions.
Also Blur does not only reduce the damage you take, it stops you from being effected by hits and those effects are even more dangerous. I am talking about things like being engulfed by a Shambling Mound or Being Swallowed by a Purple Worm or even being poisoned by a lowly Imp. Those kinds of effects are devestating in melee, but they require a hit to land.