Back to what I really want to know:
Your evaluation seems disappointed the Bladesinger can't emulate your image of an "elf with a sword in one hand, spells casting from the other, all while wearing some cool elven chain".
But what do you really feel is missing? What martial offense would you give the class?
I guess if I were to design the class option myself, I would have given it less defense and more offense. There needs to be some offensive ability at the 2nd level bladesong abilities.
I'm not a game designer, but since you put me on the spot, I guess I would remove the 2nd level AC bonus, and instead provide the Int damage bonus. Wizards have other defensive options anyways, and the Bladesinger gets the ability to absorb damage later on as well. At 14th level I would then apply the AC bonus.
I'm asking because you never discuss why it might be that the SCAG class can't fulfil your Bladesinger hopes.
It's the lack of offense, or to be more specific, it's the concentration of lower level bladesinger powers on defense. If I'm going to swing a sword in combat with my wizard, the sword needs to be a solid offensive option for the character.
You say the traditional Bladesinger is "Bad or nearly useless", which is somewhat strong to say about a full spellcasting class with access to level 9 spells and the full Wizard spell list.
Let me qualify that criticism. The Bladesinger becomes bad as a melee combatant. Naturally, any full class Wizard is quite powerful when they cast spells. Note that I gave the class a green rating if just used as a defensive buff for a regular spellcaster.
However, if that bladesinger draws their sword and becomes worse for doing so, I don't think the option achieves what it should.
Is it the cantrips? Would your evaluation of the traditional Bladesinger be better if the new cantrips simply didn't exist?
I won't say they don't make any difference at all, because when evaluating character options, I tend to evaluate them compared to similar options. I couldn't help but note that certain other melee/spellcasting options become more powerful with the addition of the cantrips, which makes the Bladesinger worse by comparison.
However, since there are absolutely zero offensive abilities this class gains (beyond that of a normal wizard with a sword) before level 6, and then only a single extra attack until level 14, I would still find the Bladesinger lacking as a melee option, compared to say a Valor Bard or Cleric until level 14, which seems really late to me to be catching up.
You say "this is a decent tradition for Wizards who have no intention of ever using a weapon" as if that's a bad thing.
Nope, I think it's a good thing, and a saving grace of the subclass. I could see playing the subclass just to get all the great abilities added to my wizard. I just wouldn't be drawing a sword. From a mechanical point that's fine, as the subclass offers enough power to be a viable option, it just doesn't turn your wizard into an effective melee character IMO.
But how do you incentivize the Wizard to ever use a weapon, given all her spells?
If you take a look at most Clerics, you will find them using weapons, though not all Clerics use weapons. Either choice works well depending on your build/focus. This is what the Bladesinger should achieve, to have drawing a weapon not be a bad idea. I note that Clerics have a number of spells that help them be more effective in melee (Spirit guardians and Spiritual weapon come to mind), perhaps Bladesingers should be given abilities or spell options that provide benefits that compare in offensive boost.
And again I ask your comment on the argument the Bladesinger turns the blade-wielding Wizard from terrible to not-an-embarrasment-to-yourself which might be good enough.
Yes, I do think with a bit of a Multiclass you can still make the concept work, which is definitely a good thing. I note Mellowred made a suggestion regarding Paladin 2/Wizard X, I hadn't thought of that, and the more I think about it, the more I like it. In fact, I think I'm going to give him xp for that post right now.
If anything, your assessment of the God Wizard build makes me weep for the Sorcerer. One of the last small reasons to ever play a Sorcerer was the extra sturdiness of the Draconic subclass. Not any more.
In my opinion, the draw of the sorcerer has nothing to do with sturdiness, it's about those sorcery points and how they can enhance spells.
Hopefully that answers your questions.