If the bladesinger is meant to be balanced with other traditions and the EK with other archetypes, then, if we presuppose that the wizard is superior to the fighter, it would make sense for the bladesinger to be able to appropriate quite a lot of fighter-ness without overshadowing the other traditions, while if the EK appropriated much wizard-ness it would quickly stand out as superior to the other archetypes.
So, we've slid from 'fighter and wizard are otherwise equal' to 'traditions are equal and archetypes are equal'. Not a problem, just want to make sure we're on the same set of goalposts.
That agreed, your argument still doesn't logically follow, largely because of the reasons for the overshadowing. Wizards overshadow fighters because wizards contribute strongly to all pillars of the game while fighters only contribute strongly to the combat pillar of the game. When considering what the Bladesinger offers compared to other traditions, it doesn't increase ability in the social or exploration pillars, and, arguably, decreases effectiveness in those pillars if it's core concept is applied (ie, fighting in melee) by siphoning resources to bolster the melee strength (spells to enhance survivability and effectiveness in melee). For this it gains some moderate effectiveness in melee, an area other wizards are very weak in, but does so at increased risk. Since wizards can be just as effective in combat without engaging in melee, the Bladesinger really only offers a novelty exchange to allow a wizard to be in melee, but doesn't increase effectiveness at all.
Compare this to the EK. The EK gains 1/3 progression wizard casting, which at first blush would seem to enhance the fighter's ability in other pillars and make it a stronger subclass. But it doesn't do this at all. The spell selection is tightly limited to abjuration and invocation, both of which are combat pillar spells. The few the EK gets outside of those two schools aren't significant in expanding their effectiveness outside of the combat pillar. And, for those spells, they don't significantly enhance combat pillar effectiveness over the other archetypes.
The EK getting wizard spells doesn't mean the EK is able to assume the abilities the abilities of the wizard class -- like the other fighter archetypes, the abilities provided by the EK archetype supplement it's contribution to the combat pillar in a roughly equal way. The Bladesinger tradition abilities, conversely, offer a choice - be a mediocre fighter with good defense or be a wizard - and that choice isn't multiplicative, it's either or. This is why the Bladesinger isn't an overpowered tradition - it can't do both at the same time, and the trade off, while versatile, means it's not as effective at either.
So, the wizard can overshadow the fighter because it contributes strongly to all pillars while the fighter does combat only. The Bladesinger just offers the option to contribute to the combat pillar as a wizard or a mediocre fighter, but not both and not in a multiplicative or even additive way. The EK archetype doesn't let a fighter become a wizard, but instead narrowly focuses, like the other archetypes, on enhancements to combat pillar contribution. The Bladesinger overshadows fighters (including the EK) because it's still a wizard, not because it can sometimes be a mediocre fighter. It doesn't overshadow the other traditions because it doesn't offer any increase in power, just a bit of role versatility. The EK doesn't overshadow other archetypes because it doesn't gain abilities in other pillars and the abilities it gets in the combat pillar are on par in effectiveness as the other archetypes. The reading that since the EK casts spells as a wizard that it becomes better than other archetypes misses the fact that it's not casting that makes wizards overshadow fighters, it's what and how many things the wizard can cast.
