I notice that you didn't address any of the counter-examples I gave with magic items that cast spells using an action where the spell does not take an action to cast (either more or less). I'll give the benefit of the doubt that you just wanted to focus on other points, but if you continue to argue that will need to be addressed because it looks like we have a clear counter-example in the rules that doesn't support your interpretation.
I actually missed that. I have ADHD and sometimes I focus on some things and miss other things. It wasn't intentional.
If the magic items allow that, it's not a counter example. It's an example of specific beats general. Those items specifically allow the casting of that specific spell as an action. The Bladesinger ability does not call out Mending as being castable with the ability or reducing longer casting times to 1 action, so it doesn't.
It makes the casting time of the cantrip irrelevant. It replaces it with an attack. Explicitly.
No. It explicitly allows you to put a cantrip into the attack. It does not explicitly reduce the casting time, therefore it doesn't.
Again, see the example of See Invisibility allowing you to see the invisible person as if they were visible, but failing to remove the disadvantage for being invisible because it doesn't specifically do so.
I didn't ignore your magic item example, but it you seem to keep ignoring that in 5e specific means very specifically called out by name or it doesn't happen.
There is zero support in the rules that the casting time of the cantrip matters for this replacement.
Other than specific beats general anyway, which requires the casting time to be specifically reduced or it doesn't reduce it.
Because nothing needs to. The casting time is transmuted to "one attack".
No. That is not specifically said, so it does not do that. Only what is specifically said happens by RAW.
But since you DO NOT HAVE AN ACTION FREE to cast a cantrip because you have already spent that action on the Attack, then by this reading you can't cast any of the 1 action cantrips. If the casting time is not changed, you don't have the action left to cast it, even if it would happen to take the same amount of time as your Attack action. If this is correct it will stop all of the cantrips except the ones like Shillelagh if you still have a bonus action.
You don't need an action free. The ability specifically allows you to put the cantrip into the 1 action length attack and use it, but does not specifically reduce the casting time, so if the time is longer than the 1 action attack, it can't be used. Casting times of bonus action and action can fit within the 1 action length attack just fine.
So this doesn't support your point at all. EITHER it's true, and you can't cast any of the 1 action cantrips, or it's unrelated and the feature does exactly what it says - allows you to cast a cantrip with the casting time of "one attack".
This doesn't make any sense. The cantrip 1 action length of water can fit just fine in the 1 action cup of the attack action. However the 10 round(10 actions, excepting specific cases of additional actions) casting length of Mending does not fit inside the 1 action cup of the attack action.