It would only be one BB/GFB, not two (two attacks, but only one is a cantrip), unless there is some other shenanigan beyond bladesinging I'm not familiar with. And BB/GFB have anti-synergy with GWM, like any other damage increase without an increase in accuracy. But yea, full Dex bladesinger (only need Int for the AC boost while bladesinging) would be pretty beastly. Restrict it to Int for attacks, though, and I don't think it's too bad.
At 6th level:
attack: 2d6(7)+3dex+10 gwm = 20
cantrip: 2d6(7)+3+10+1d8fire(4.5) and 2d8(9) fire+3dex (second target)= 24.5 1st target, 12 2nd target
56.5DPR at 6th level without using spell slots, while also getting a third attack for 20DPR more as a bonus action every time you crit or drop an enemy.
That is compared to the current 6th level bladesinger that maxes dexterity and has a 16 int and does 33.5 (21.5/12) with no bonus action attack.
Intelligence only would not be much better. It would require another feat or a race to get medium armor, but you could still be rocking 57DPR by 8th level with an extra bonus action attack every time you crit or dropped an enemy. And it gets worse at higher levels because the intelligence stacks on the weapon damage, stacks again on the cantrip damage and can be used on spell saves.
Bladesinger is already a very powerful subclass and this would take them from MAD to SAD.
Those are the kind of builds I want to encourage!
Why even have strength as an ability if you can get everything you can get with strength from another stat? The ONLY thing at all strength would be good for is carrying capacity and grappling/shoving.
DIfferent groups, then. 14 Con is pretty much the standard, and 16/16 or 16/14 in main/secondary is also easily the most common distribution.
A lot of people here say that, so you are not unique. This is the real reason you are running around with a bunch of 8 intelligence and charisma fighters. If you invest in a character only to be good in combat - if you make him a ham-fisted bag of hps, he should not be good out of combat. I don't think you should fix this by letting them dump strength. I think you fix this by encouraging players to invest in other stats instead of Constitution.
If you make Constitution 10 instead of 14 you have 5 points to spend on other abilities and a much more playable character.
Your experience is not unique, but I think Constitution is overrated by many on this board. At 9th level a constitution 2 points higher means you will survive a whopping one more hit from a CR1 Orc than you would otherwise survive.
I mean, I can definitely see it for gishes, I'm not sure if other classes would bother. Why does a non-bladesinger wizard need to boost their weapon attack? Why would a barbarian bother to not use Strength? Why would a fighter?
There is simply no reason for a fighter to invest in strength. Why would a fighter bother to use strength when they could dump it and instead attack with a stat that is FAR better both for saves and for skills?
A fighter with custom lineage could start with a 17 wisdom enabling +3 to hit/damage at 1st level with a greatsword. He would also have a +3 wisdom save, a passive perception of 15, and a +3 in the 4 other wisdom-based skills (without proficiency) while still doing the most damage and having as many hps as any other fighter build at this level. Bump wisdom to 18 at 4th level with a half-feat like Fey touched, skill expert or resilient and it makes him even better at wisdom saves, better at 5+ different skills and boosts hit/damage again. Further aside from this he could go EK and use wisdom instead of intelligence for casting purposes!
With a Barbarian it would depend on if you allow rage damage and reckless attack on a wisdom-based attack. If you do he is going to dump strength too for a different ability (most likely wisdom). If you don't, no one who understands numbers will play a Barbarian when there are better classes available (virtually all of them).