A background for +2 int, +1 con, magic initiate is not a poor choice for a bladesinger. It's one of the best.
It is weak if you want to melee and it is the equivalent of picking an Eldritch Knight background that does not boost strength and then Fighting with a Strength weapon.
We are talking about a melee character here, Dexterity is their primary ability for that.
We can quibble over whether it's the absolute best or not, but it's one of the best.
No it is not. It is a weak choice if you want to play a melee Bladesinger. There are at least half a dozen better choices. If you want to play a backline caster with some defensive options that is a different story.
How about this - limit the Eldritch Knight to a dagger, after all that is a great choice and you can get nick with it! Why don't we use that to compare numbers?
I'd vote magic initiate as being better.
For a melee Wizard it is not better than Alert. For a melee Wizard, the only PHB Origin feat that competes with Alert is Lucky.
Moreover Magic-Initate (Wizard) is the worst available Magic Intiate feat.
None of my choices were bad.
The background makes for a weaker character.
And I assumed a single cast of false life for my single encounter 4 round adventuring day. A level 3 slot since you don't have an extra 4th till you short rest.
You will cast False Life after every battle where you lose your temporary hit points as long as you have spell slots left. Assuming a "single cast" is like assuming a single use of Second Wind.
Arcane Vigor is a good call out. I'm surprised you are just now bringing it up. That's one I can easily add into the rotation.
Arcane vigor is not going to be cast a lot, but it is there if you want to use it in combat. Unless you are really getting beat up you wouldn't use it at all though.
This wasn't true. The Fighter also can cast False life in a level 2 slot and has an hp lead due to d10 vs d6 hp.
The bladesinger has every spell slot the fighter has plus 6-10 more. So your hypothetical fighter is entering 1 combat with 11 more hit points and every other combat with less hit points (assuming he gets hit)
I labeled them. 4x shield. 1x magic weapon. 1x False life (level 2).
And for comparison the Bladesinger has all those plus 2 more False Life, and I don't know lets just say an additional 2 Jump, 1 Silvery Barbs, 2 Absorb Elements, 1 Longstrider, 1 misty step and 1 Backlash.
All of these are going to improve the Bladesinger's effectivess in combat and none of them are going to compete for an action.
I played TO level 20 as in every level from level 1 to level 20.