I've only played a bladesinger and then only up to level 5 so I don't have any higher level experience.
I've noticed that while I am hard to hit in melee, my low hit points means that saving throws I fail (or succeed on for half damage) are still quite punishing. Greenflame blade really helps my damage in melee, even if there wasn't a secondary target. Having an additional defensive spell cast before combat really helps (I went with blur). The +Intelligence to concentration checks is nice, unless like me you have a tendency to low roll your concentration checks.
Bladesong is a great ability, however, you can only maintain it twice/short rest for 1 minute, and if you are incapacitated you lose it along with its defensive benefits (I think once I got hit with a hold person, losing it).
There have been times where it was a better decision for me to cast haste on the party monk and then hang back and blast with fire bolt or drop a fireball instead of engaging in melee.
The benefits of the bladesinger I found was the versatility of being able to engage in melee or being able to hang back and blast with spells, it was very easy to swap between the two stances depending on the situation. Often I would activate bladesong, move 40 ft (due to the bonus from bladesong), blast an enemy with firebolt, then engage them in melee the next round. Having some close combat spells like burning hands or thunderwave is also handy.