To make a less offhand remark, I think the strength of Bladesingers is the ability to contribute to a parties combat strength AFTER all of their spells are expended. Before that point, they should be played as a wizard and stay out of melee at all!
After the spells are fired off however, when a "normal" wizard would be stuck with just slinging cantrips, they are cantrips + melee at the same time.
So I think the question is really "What's your best melee option after all your spell slots are used."
Having recently played a bladesinger from 1-13, I can tell your premise is flawed. Not that you can't play them that way, but that is not the best way to do so. It is distinctly inferior.
The idea is to get as much out of your slots as possible, using your weapon abilities to sub in for cantrips and other "low return" actions during the adventuring day to bring up your average
effectiveness per action. It's why short days where you can just Nova favor casters - they have a very high effectiveness per action because they fill each action with a high return.
Think of these two cases:
CASE A: Only use slots for all encounters until out of them, and then use unbuffed weapon attacks.
CASE B: Mix using spells and weapons as appropriate for each. Times it's a big area of effect or crowd control, other times it may be a multi-person buff like an upcast fly. Some combats may be winding down and it's saving slots for later combats by using weapon. Other combats may be against foes that are resistant to damage types you have, have Magic Resistance giving advantage on all saves, or other reason why targeting them with spells is less than useful. Many combats do not need lots of slot use and a single long-duration Concentration buff works out to be the most efficient in terms of a good per-action efficiency now and additional slot resources when needed later.
Since, if it's the most efficient for that adventuring day, CASE B could mimic CASE A, we push forth that there is NO SITUATION where running a Bladesinger with CASE A is the superior choice. Your option is
never the best one - it can't be. At the very best it can tie actually thinking and using your abilities where and when they make the best sense.