I've seen them played by people who want to gish - either pure or as part of a multiclass though even then it's usually just a dip elsewhere - they need INT and that doesn't have a lot of synergies. Booming Blade / Green Flade Blade, and once they get Extra Attack I see Shadowblade or a buff like Haste or even better Greater Invisibility, which helps keep away the damage that can break concentration.
Often goes ranged with elven longbow prof when low on HPs ... until they hit the levels where cantrips outdamage two attacks. And in that time some discover they can be just a normal wizard but super defensive.
I've seen a player start with the bladesinger as wizard+defense+concentration, never really weapon wielding except the occasional opportunity attack, or when it's better than a cantrip. They were just playing as a pure wizard with defensive options.
From what I've seen (and some has been at other tables at my FLGS/friends stories so not direct observation), they shine as weapon-user when the DM does the 6-8 encounters a day and gives out a decent number of short rests. Because spending actions on weapons is an action not spent using a slot, and that means they have plenty for when they do cast.