That is a completely absurd claim.
First, Bladesingers won’t reliably have Bladesong up in any campaign I’ve been in, until around 11th level, when most campaigns are done or ending. At low level, they have it 2/day.
If you follow the guidance for 6 fights a day (and few do IME) they will be in bladesong fully half of the battles they are in at 5th level and more than that at high level.
At levels 2-4 ONLY they have it twice a day. That is one sixth of the possible bladesinger levels. If you cap it at level 12 like many campaigns it is still only one third of the levels they will play as a bladesinger. During most of the game they will have it more than twice a day
Secondly, Shield costs a spell slot and lasts at most 1 full round. So no, their AC isn’t reliably that high. And I don’t know how you figure attacks against them will reliably be at disadvantage. You’re gonna need to back that one up.
Protection from good and evil or blur will give opponents disadvantage (PGE certain types of opponents only) and every bladesinger I have seen played by both me and every other player had these spells and they were the most common leveled spells cast other than shield. At levels 2-5 it is some battles, but by 6th level it is just about every battle they are in.
Shield is only used when their already high AC is hit (typically at disadvantage) and they rarely run out.
Third, they still have garbage HP, lower attack bonus than the Druid, and no particular defense against things like Call Lightning except mitigation via more spell slots, which Druid also has, and the BS has no self healing.
Call lighting is not "melee" combat. I said a bladesinger would beat a Druid in MELEE at any level.
Since you brought it up though, the math does not support this statement.
A 6th-level Druid with an 18 W who casts call lightning on a bladesinger with an 18 Dex is going to typically do 6 DPR (8 DPR with no save or or 4 with one an an 11 needed to save). When the bladesinger hits him the next turn, the druid himself is going to take an additional 3.5 lightning damage or up to 10.5 if she chooses to use a higher level slot to absorb it. This in addition to the 20+ in weapon and thunder damage he is going to take with 2 hits.
And the Druid doesn’t need to go MAD as badly as the Bladesinger does, so the Druid is going to resist things like Hold Person more easily.
Again Hold Person is not MELEE. That said counterspell is more effective against hold person than a 20 wisdom is.
Any Druid subclass could gank a Bladesinger, unless the BS hides behind some summons and acts like a normal Wizard, in which case they’d be better off as an Abjuration Wizard. The only subclasses BS can stand up to reliably are the melee subclasses, and the Druid is still gonna win 7 or more times out of 10.
The discussion was about MELEE but even a Druid who resorts to spells is going to typically be on the short end of the stick because the wizard has better spells.
The BS if fun. It’s not in the top tier of any category but AC, though, and even that is situational.
In melee AC is extremely important. That said, due to their unique extra attack they are pretty darn high in damage as well and above 10th level they can take more damage over the course of a battle the vast majority of classes. Now there are a lot of caveats there, but that is the math.
Now, MC builds using Bladesinger can’t be really powerful in melee, as long as you don’t mind losing out on greater magical power overall, (and I wouldn’t. I signed up to play a gish, not a mage), but then so can Druid MCs like taking Beastmaster Ranger (with Tasha’s options) or Monk (all your monk stuff works in wild shape RAW) or War Cleric, or nearly any Fighter.
Bladesinger is going to be better in melee if he does not MC at all because of the bladesinger extra attack, song of defense and at very high levels song of victory. These three abilities are a strong disincentive to multiclass at any phase of a bladesinger's career, even if he wants to play a pure melee style. I guess if you are playing at very high levels you could multiclass to fighter (for AS and maneuvers) after 14th level but you lose access to those very high level spells if you do that. That said this would bump your melee a little bit if that was what you were after.
In terms of damage, Bladesinger extra attack is close to 3 attacks at 6th level and because the cantrip portion it scales and stays ahead of other classes that are limited to 2 attacks with extra attacks. The extra attack gets comparatively weaker for other classes as levels progress because their attacks do not scale as well as the bladesinger extra attack (except for Fighters). Also a substantial portion of the bladesinger damage is magical, overcoming resistances. A bladesinger at 6th level can reliably deal 20.5 DPR, with no hit penalty, limited use abilities or special circumstances to land that damage. With certain stipulations met he can regularly boost that to 27-30 without tapping limited use abilities and it goes up from there at higher levels.
But single class? Druid wins.
Not in melee he doesn't