Yes, I’m aware. That’s sort of the point. The Wizard decreases offense the more they have to take attacks.
But if we are talking about tanking that is irrelevant. If your point is a wizard can't consistently be a striker or controller while being a tank, that is true but it is true for Barbarians and fighters and other tanks too, however a wizard optimized for tanking can actually switch gears and be effective at something else better than most others optimized for tanking.
In tier 2 a shield spell is worth about 20 hit points, so the four 1st level slots at 5th level are the rough equivalent of 80hps in terms of tanking. Now that is 4 magic missiles she can't use but she has a ton of other spells she can.
That is the epitome of white room thinking, though. You’re leaving out HP, con saves, punishment tactics, surviving the randomly very high attacks for big damage, incentive to attack, and opportunity cost.
No it is not white rooming. I have played multiple bladesingers and seen others play bladesingers from level 2 to level 16. It is the most popular single subclass at my table. I understand the math behind it, but what I am saying comes from the table, not from the white room.
A. HP - Hit points are a limited resource, being any wizard allows you to trade spells for hitpoints numerous ways. Aside from shield, absorb elements and a host of defensive spells; one false life cast at 5th level will give a 9th level wizard more hps than an 8th level barbarian and she is getting hit A LOT less than a barbarian. On top of this, at 10th level being a bladesinger specifically allows you to trade spell slots for hit points directly as a reaction for damage you can't counter through another method.
B. con saves - There are several mechanics at play here.
1. AC: Most of the things that make you do a concentration save or a constitution save for poision etc are damage you take from an attack. This is happening far less frequently on a bladesinger (even out of bladesong). So your saves are not good or perhaps even bad but you don't have to save against attacks nearly as much as other casters. Making 1 save with no bonus is generally easier than making 3 saves with a +5.
2. Bladesong: For concentration specifically, while in bladesong you will have an intelligence bonus. You will typically be in bladesong in about half of your fights in tier 2 and usually the most pressing fights. Now this is only for concentration, so it is only a part of your constitution save concern but it is the most important part for a caster.
3. Absorb elements/dexterity: Most of the things that are not attacks and will make you save can be mitigated/reduced by absorb elements and your high dexterity.
The combination of these three things make a bladesinger pretty darn good at avoiding failed constitution saves as far as casters go, IME in general better even than sorcerers who are proficient.
C. Randomly very high attacks for big damage - traps are a big problem. At our table we had a bladesinger (the legendary Afalie) that did not get hit one time in combat at all for several entire levels (I think 4-7 or something like that) but that character did go down from a trap during that time. The bladesinger is a tank but she should not be the girl scouting ahead or opening chests. In combat this is not usually an issue. It is possible to get downed out of the blue, but the AC, defensive spells and reactions available make bladesingers less vulnerable to this than most other classes, including most martials, despite their lower base hit points.
For example, when that ancient Red dragon breathes on you at 10th level, that 91 points of dragon's breath will never kill a 10-constitution 11th-level wizard (46hp). She is going to cut it in half for 45, or if she is a bladesinger and gets really lucky and makes her save she can drop it all the way to 15 damage by spending a 6th-level slot. An 11th level fighter with lower than a 14 constitution will be downed and at 14 he is going to be at 1 hp unless he saves.
D. Incentive to attack - yeah this is a problem. Enemies want to attack others, but this is an issue for any tank. Enemies want to attack your squishy characters, you counter this the same as with any other tank, try to make them pay for attacking others.
E. Opportunity cost - Again valid criticism, but one that applies to any tank. Putting a shield on your fighter is a big opportunity cost versus a greatsword and GWM or a halberd and PAM. What makes the bladesinger better is she can switch back and forth easier than say a fighter. A bladesinger is going to have a ton of defense prepared and that comes at a cost, but you generally still have a few offensive spells ready to go to, where the fighter who did not take GWM can't decide to drop the shield and go full on offense in one fight when needed.
How does the WF or Gith Wizard hit 17?
Gith with Half plate and dexterity.
Warforged: Mage armor 13+ 3 dex+1 warforged =17
That assumes of course that we are limiting characters to armors they are proficient with.