First, cantrips that have an attack are still "casting a spell" that's why you can't have an eldritch knight use Booming Blade on Four attacks.
The Bladesinger can cast a cantrip as part of the Attack action, instead of the Cast a Spell action.
Second... I likely would be highly suspect of allowing the attack from Haste to trigger two-weapon fighting. But, considering that looking over the list of what actions are allowed under your hasted action, the only two things you can do with your main action that you can't do with the Hasted action seem to be dodge and casting a spell. So in the vast majority of cases you have already made an attack with your main action, or you could do whatever you wanted to do with your main action as your haste action, so it won't matter.
It works RAW, same as this combo. Two-Weapon Fighting requires you to take the attack action, so if you do that with your Hasted action and Cast a Spell with your first action, you still get a bonus action attack if you are dual wielding.
And finally, this is requiring a lot of steps. First off that the Extra Attack Feature is possible to activate during Haste. Second that the ability to swap one attack for one cantrip can be done multiple times a turn, because you already swapped one attack for one cantrip, and that you can swap your only attack for a cantrip.
Even if RAW, I think that is too much to be said to be RAI.
We currently have no comment from Jeremy Crawford on whether this was RAI, so there's no reason to suspect that this was or was not RAI. I don't think they specifically were trying to allow this combo RAW, because the game designers aren't very good at finding exploits in the game's mechanics, but unless JC comes and explicitly says "you should not allow this combo at your game" or they make an errata or Sage Advice to get rid of this combination, I'll allow it. It's RAW, after all, and requires you to concentrate on a spell.
For example, if an attack as the grapple rider... it is part of that attack. That is like you trying to argue you can cast spells as part of your attack because Hunter's Mark damage is part of your attack. You are talking about using an ability that is meant for "Instead of making two attacks, make one attack and one cantrip" and letting it say "instead of making an attack, you can cast a spell when you normally cannot cast a spell"
It's not the same thing, as there is no feature that allows you to do Hunter's Mark as part of the same attack, and doesn't grant you extra actions. Extra Attack grants you an extra action, which you can use to Attack, while the Bladesinger's Extra Attack lets them replace an attack with a cantrip. You still have to fulfill the requirement of "one weapon attack" from Haste, which is why you're restricted to GFB and BB. It's really very simple.
Haste does not let you make a second attack with Vampiric Touch, a melee spell attack granted to you as part of the spell. Haste does not let you move Moonbeam twice. Haste does not let you use sunbeam twice. Haste does not let you use Call Lightning twice.
Because those would be actions not allowed in the spell's description. A cantrip cast as part of the Attack action is the Attack action (which is allowed by Haste), not the Casting a Spell action.
Haste does let you use Shadow Blade twice. Why? Because Shadowblade creates a weapon you wield to make attacks with, it is not a spell attack.
(As long as an ally has the Haste up, that is.)
That is correct, and it is because Shadow Blade lets you take the Attack action with it (unlike Flame Blade, for some unknown and probably idiotic reason). Doesn't have much to do with the current discussion, as we all know that the Attack action is not the same as the Cast a Spell action.
Casting a spell is not something you can do with your hasted action. And in the case where you have a single attack, I do not believe the intent is for Bladesingers Extra Attack feature to activate.
Specific beats general. The general rule is that you can't take the Cast a Spell action from the Hasted action, but the Hasted action allows you to Attack. The BS's EA lets you replace an attack with a cantrip as part of the Attack action, so the combo works. The key fits the whole.
I don't believe they designed the Bladesinger's Extra Attack feature specifically to allow this combo, but as they currently have said nothing on the matter, there isn't any sign that they do not want this combination to be viable.