Something's up here. In my hard copy PHB and on Fantasy Grounds and on 5e SRD, a 6th level Wizard has 4 1st level slots, 3 2nd level and 3 3rd. 4/3/3. Arcane Tradition then recovers 3 levels of slots.
Nope, my bad, scanned the wrong line.
She doesn't need to. The cleric uses a Prayer of Healing (2nd level) to bring everyone back up to nearly full, and she spends 1 Hit Die. This is more efficient than the alternatives.
BS is on 2/2/3 casts + 3 levels of slots
Cleric is on 1/1/3 casts + channel divinity
So, after the second fight, same as the first, the cleric is using a 3rd slot for the healing word and is down to 0/0/2, plus the prayer so 0/0/1. The cleric has burned their entire spell list to prop up the bladesinger, and you think this means the bladesinger is a really good tank?
So you are anticipating three super-deadly encounters in the same day? Please play these and see what is left of the party that doesn't have the BS. Cleric is happy to use their 4 qualifying slots for Warding because that's really the most efficient thing to do.
Huh? BS could use other slots because BS is presumably not tanking six super-deadly encounters a day! Most encounters don't merit the cast of a single Shield, nor a Warding Bond. These Giants are +8 attack, dealing 3d8+5 with Greatclub, or 3d10+5 with a rock.
3, 3 encounters is what I said, not 6. Inflating my statements to win arguments doesn't reflect well on you.
And, huh? Most encounters don't merit the casting of a single shield? What numbers are you running to show this? Certainly not the one you proposed, and are you really going to play the bladesinger on the assumption she'll never need to cast shield so you can expend those slots freely on other things?
Bladesinger, AC 23, standing up to 3 giants at tank. Assume the bladesinger is out front and the giants are in 'smash closest thing first' mode to start. All three giants attack the bladesinger, first with a toss of a rock, then in melee second round --
Bladesinger wins initiative 75% of the time (+4 vs -1).
Round the first:
Bladesinger casts blur, moves to intercept (or just better position to protect the party). Activates bladesong.
Giants all toss rocks at bladesinger and close. Hit on 15+, or 30%, blur reduces to 9%, crits are 0.25%, 3 attacks, so bladesinger is hit by 1 rock is roughly 25%. One rock deals an average of 21.5 damage. Bladesinger at 6th with 14 CON has 31. Probably wants to shield that.
Chance 2 rocks hit is ~1.5%. That's dead, so the bladesinger is definitely burning a shield here.
Chance all three rocks hit is 0.07%, so better chance of a crit at 0.25%. Bladesinger can't shield that, and just goes down.
So, first round is ~26.5% chance of needing a shield and 0.75% chance of death.
Round the second:
Bladesinger now in melee. Does some GFB for some damage.
Giants all attack bladesinger. 6 attacks, now at 9% each, with 0.25% chance of crit, doing an average of 18.5 damage.
Chance bladesinger is hit by at least:
1 attack -- 43%
2 attacks -- 3.4%
3 attack -- 0.25% (this is close to crit chance, we we'll ignore below this)
So, that's a better than 45% chance the bladesinger will need to shield in the second round. Added to the first round, the averages say the bladesinger will need to shield at least once in this kind of combat. And that's with blur up.
The chance to crit for the melee attacks, by the way, is about 1.4%. Even with blur, that's not negligible, and it's a 1-shot down for the bladesinger.
So, the bladesinger has to keep that slot available and should expect to expend at least 1 in that fight.
Now, if the bladesinger loses initiative, things are very, very bad for the singer.
Round the first --
3 rocks at AC 18, no blur, hit on a 10+ (55%) crit on a 20 (5%), damage 21.5 or 37.5 on a crit. Bladesinger sucks up
1 hit: 91% chance
2 hits: 43% chance
3 hits: 17% chance
So, that's 91% chance of half the hit points or a shield use, a 43% chance of dead or less than half hp (1 shield) or 2 shields, and a 17% chance that you just burned all of your 1st level slots on shield. That's some bad resource usage!
A crit ends the discussion, and there's a 14% chance of that happening in the opening round.
So, sure, with preparation and resource usage, the bladesinger can tank, but her shallow hitpoint pool means she needs to keep slots open to shield for bad luck. Without preparation, the bladesinger goes down as fast as a normal wizard.
You can count on winning initiative 75% of the time for staying up, but you might want to save those slots, just in case.