You can't visually tell a difference by the effects that are produced, but there is a mechanical difference. They're different actions. If there was a feature for a rogue that lets them replace their attack with a Dodge action, but keeps it a part of the Attack action, when they Dodge this way they will actually be taking the Attack action, even though there is no visual difference between the effects that are produced.
No obvious difference, but there is a difference in the type of Action you take. 99 times out of 100 it will not matter, but it matters in this case.
The problem with that Rogue example is fairly obvious. They dodge, then attack with their bonus action because they made an attack. But they did not make an attack, they dodged.
The only reason the Bladesinger can replace an attack with a cantrip is in the context of making two attacks, because of the known problem with two attacks being better than a blade cantrip. It is not mean to allow you to cast a spell as a hasted action, which is what you are doing.
If you read the feature as "one weapon attack only", where the only refers to the quality of the attack, you can't add Divine Smite, because that is doing more than just making a melee weapon attack.
Additionally, Haste's Attack action requirement never states "Features with the name Extra Attack can not be used with this action", it very explicitly states "One weapon attack only". Whether or not the ability to replace an attack with a cantrip is part of the Extra Attack feature does not matter, as the writing of the Extra Attack feature does not make that fact matter.
Did you just ignore the parts I bolded for the Bladesinger? Because they are what matters for that section. I'll show you again.
"You can attack twice, instead of once, whenever you take the Attack action on your turn.
Moreover, you can cast one of your cantrips
in place of one of those attacks."
Morever -> As in as an additional part of this feature
in place of on of those attacks -> Specifically referencing making two attacks with the Extra Attack feature, as laid out in the first sentence.
This does not allow you to replace any melee weapon attack with a cantrip. If it did, you could replace your bonus action Attack Action with a cantrip. Because that is also an Attack, just as a bonus action and with special rules. After all, you can Divine Smite or Stunning Strike a Bonus Action Attack, so then this would work, right?
Or, here's another thing. You can Divine Smite on an Attack of Opportunity, can you cast Booming Blade as the same? No, because then you would be getting the benefits of War Caster for free, and this doesn't allow for that.
This is what you are missing. You are reading the Bladesingers ability in a way that allows them to replace any attack they make with a cantrip. That is not what the ability does. It allows them to replace a single weapon attack they make with their Extra Attack feature with a Cantrip, not any weapon attack they make under any circumstance.
Absolutely not by any reading of the rules. That's a strawman argument if I've ever saw one. A bonus action attack from Two Weapon Fighting is not an attack action. If it was written the same way as Haste, that combination would work, but it is not written that way. If Haste was written the same way as Two Weapon Fighting, this combo would not be legal.
A Two-Weapon Fighting bonus action attack is not the Attack action, while the Hasted attack is written as part of the Attack action.
See, here is the problem though, you are saying that the Attack Action is different than an Attack Bonus Action and an Attack Reaction, but that is not how the game works.
If a Monk uses Patient Defense, they take the Dodge Action, but they do so as a bonus action. If they are a dwarf and they have Dwarven Fortitude they get to roll a Hit Dice and heal. That feat says "when you take the Dodge Action" and you did, you just did so as a bonus action.
Because the action "dodge" is the same action whether you do it as an Action, a Bonus Action, or a Reaction.
Let us say there is a theoritical item that gives someone +2 when you use the Help Action to aid them. If it was on a Mastermind Rogue who used their special ability to use Help as a Bonus Action, would the creature get the +2? Or would you say they took the Bonus Action Help, and therefore it wouldn't qualify because it needs to be the Action Help and those are two different things?
I certainly don't think so.
So, if you can replace any attack of a Bladesinger with a Cantrip, then you could make bonus action cantrips and reaction cantrips. But you can't, because the ability is specifically meant to be used with their Extra attack feature. Nothing else. Unlike abilities like Divine Smite, which work for any attack you make, regardless of how you made them.
GFB literally requires you to make a weapon attack, so it wouldn't be banned. Grapples and Shoves aren't melee weapon attacks, they're special melee attacks, and unarmed strikes are melee weapon attacks, so they would be allowed.
Green Flame Blade is casting a spell. It is literally the "Cast a Spell" action. Just because you make a melee weapon attack as part of the spell does not make it an Attack Action.
We can pretty much prove it too. If you are counterspelled, your action fails. You don't still get to make a melee weapon attack despite being counterspelled, because Counter Spell cancels the entire text of the spell, including the part about making an attack roll, so you get to do nothing.