Probably because prime casters are intended to suffer a slight penalty to their basic attack routine damage (cantrips vs melee attack routines) in exchange for their versatility. The Bladesinger remains a prime caster whereas the Eldritch Knight remains a fighter.
I'm going to start with a disclaimer, because the rest of my post is going to sound like I think the Bladesinger should be more powerful, but understand I am talking about offensive capability. They gave the Bladesinger lots of great abilities, my concern is not that they didn't give the bladesinger enough, my concern is that they skewed the abilities towards defensive abilities, and all but ignored providing offensive ones. IMO the bladesinger becomes not offensively potent enough and TOO defensively potent to properly represent the concept of a bladesinger.
I agree with your premise, but I don't think the actuality ends up being a "slight" penalty.
Let's first look at level 3 and compare the offensive capabilities of the EK and the Bladesinger. I will only list capabilities the other does not receive.
Eldritch Knight
1) Fighting style
2) Ability to use two handed weapons
3) Action surge
Unlikely any self buff at this level will be offensive
The Bladesinger gets
Possible offensive self buff: magic weapon spell (+1 to hit/damage)
This is a pretty big divide. Let's skip ahead to level 11, does the bladesinger close the (rather large) gap? I will only list offensive capabilities gained since level 3
Eldritch Knight
1) War casting
2) Extra feat or ability increase
3) 1 extra attack over bladesinger
4) Ability to attack in same round as using cantrip
5) Saving throw debuff on creatures hit with attacks
Possible offensive self buff: Magic weapon (+1 to hit/damage)
Bladesinger
Possible offensive self buff: Haste (extra action/round)
The gap did not decrease, it increased. Consider that Haste will probably give these characters the same # of attacks per round, but the EK has all those other offensive abilities stacked on top of those attacks and the bladesinger actually had to give up his one offensive boost (magic weapon) in order to concentrate on Haste.
I will note that this is a smaller problem with other primary casters who can do melee like Clerics, because the cleric spell list is more centered on the concept of the caster entering melee so you see some show-stopper spells like Spirit Guardians (BTW was just thinking what a combo Spirit Guardians and Booming Blade would be...) and Spiritual Weapon. Wizards simply don't have a large number of spells that buff melee capability. Haste is probably going to be your best option right through most of your career.
The divide in melee offensive capability should exist, I just think it's too wide a gap.