Along with upcast false life it did save my bladesinger (a different one) from Behir Lightning in Out of The Abyss. Barely, but it saved her.
I did not say backfielders do not get attacked, but they don't get attacked as often as people on the front line and when they do they take less damage. Straight up a backline Wizard with a 10 constitution has exactly half of the hit points of a fighter with a 14 Constitution, but I would suggest a melee fighter gets targeted about twice as much (more than that if the party is tactically sound). So all else equal a Wizard with a 10 con SHOULD last the same amount of time - has half the hit points, but gets targeted half as much
However that is not the whole story, because he has false life, he actually has more than half as many hit points, and when the wizard does get targeted he has a higher AC (including shield spell), or he can get resistance through absorb elements, or he can cancel crits with silvery barbs, and in some cases he is wearing greater invisibility or popping in and out of a fog cloud and can't even be targeted at all because the big-bad-enemy spell is one you need to SEE someone to target. So he has more than half as many total hit points, gets targeted half as much and takes less damage when he is targeted. That all adds up to goes down less often.
When I am playing a bladesinger that is how I play. I cast Fear occasionally because it is so darn good as a spell and I misty step every now and then (or I just play a Shadar Kai), but everything else as far as spells is melee stuff, usually defensive spells or haste.
I've heard this strawman thrown around before and it does not work out in play, the bladesinger is typically faster and more mobile than the enemies she is facing, it is not just disengaging, they need to go around her burning movement when they do and on her turn she is going to pass them and be on the other side of them again. Dashing around the battlefield chasing a wizard while someone wails on you is not a winning strategy .... and that is only if they can get around her at all.
Second you are not doing a lot less damage than a fighter, that was true with the SCAG bladesinger, but it is not true with Tasha's bladesinger. An 8th level Bladesinger who maxes dexterity is going to be doing 5d8+10 to anyone who ignores her and moves to go get someone else (1d8+5 attack, 2d8+5 cantrip, 2d8 movement damage). That is an average of 33 points of damage if it all hits and it is 9 more than they would take if they did not "ignore" her. That is pretty heavy melee damage at 8th level, and if you are insistent about going after another party member she can get some of that with advantage too. For comparison a fighter with dueling is doing 2d8+14 plus whatever his subclass brings to the table, a fighter with a Maul is doing 4d6+10.
Finally, your AC is so high you can use mobility to seriously disrupt the enemey. This is one big difference with a fighter, with a fighter you generally close and hope that one enemy stays there, if you have sentinel you have a feat to help you force it sometimes for one enemy, but you are screwed if it is more than one or if you miss with your sentinel AOO. With a bladesinger you attack and then move to wherever you need to be to block the enemies and get in their way. You ignore AOOs, if the bad guy takes them, then someone else can kite in and out. A hasted bladesinger typically has a move of 80 or 90! Heck if the bad guy gets close, grapple your ally, take the dash haste action move him to where you want him, then go back in and still make a cantrip attack on the bad guy.
This is aside from the fact she can do at will fire damage, at will necrotic damage and at will thunder damage, meaning she can usually find a soft spot against enemies with resistance, which the fighter can't do without magic weapons and generally can't do at all in tier 2 if he put together a GWM-PAM build or something like that to spike his DPR and actually do significantly more damage than the bladesinger .... so yeah the Trolls are going to ignore the only person that can do at will fire damage to them!