That may be so, but you said a Druid with extra attack and a fighting style would break the game in half. The bladesinger hasn't managed to do that by it's existence.
The issue with being good at melee and being good at magic at the same time is that you can mostly only do one or the other. You're not often throwing out leveled spells and making weapon attacks simultaneously. Some classes can do this, in a limited fashion (action surge, quicken spell), but if one were to staple extra attack onto the druid with no other features, it wouldn't really have a large effect, I mean, what, your shillelagh DPR doubles, but it's still less than Eldritch Blast damage, and the Druid isn't really built to wade into melee all that well using melee weapons.