Maxperson
Morkus from Orkus
Interesting. You are correct.From the PHB;
“Bonus Action
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.”
So no, nothing prevents the Bladesinger from casting 2 action spells in the same turn. You can also cast 2 action spells with Action Surge, if you have that feature.