Yes, which with the Bladesinger version of extra attack is going to…still add an attack…to what they already have, which as you’ve proposed it…would be a normal attack and a fully level-scaled cantrip. So they’d be doing two attacks and a level 5+ cantrip.
No it doesn't. Let me break this down because I think I lost you somewhere. The OP suggested that all 1st level characters get an extra attack. This would be all characters, with martials getting a 3rd attack at level 5. I suggested adding the Bladesinger version of extra attack for casters.
This would not stack all, the rules clearly state this in the PHB:
"Extra Attack
If you gain the Extra Attack class feature from more than one class, the features don’t add together. You can’t make more than two attacks with this feature unless it says you do (as the fighter’s version of Extra Attack does). Similarly, the warlock’s eldritch invocation Thirsting Blade doesn’t give you additional attacks if you also have Extra Attack."
So under this new homebrew:
1. All characters would have 2 attacks at level 1, this is homebrew
2. Martials would get a 3rd attack at level 5, this is covered in the second underlined section above, you get a 3rd attack because it says you do.
3. My further change: Casters would get the Bladesinger version extra attack at level 1. Here is the wording on that:
"Extra Attack
Starting at 6th level, 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."
All casters would get this at level 1 and they could use this or the extra attack everyone gets but not both, however there is no wording in here about getting a 3rd attack.
4. Gish characters, including Bladesingers, would get their own versions of extra attack which would all be redundant as it would be either/or since none of them specifically add a 3rd attack.
So for example if you are playing a multiclass 6th level Sword Bard/6th level Warlock with Thirsting Blade you would have all of the following options:
Homebrew extra attack which gives you extra attack at level 1 (2 attacks only)
Bladesinger extra attack which gives you extra attack at level 1 (1 attack and 1 cantrip or 2 attacks)
Sword Bard extra attack which gives you extra attack at level 6 (2 attacks only)
Thirsting Blade which gives a second attack with the pact weapon (2 attacks only)
This character could use any 4 of these options when he uses the attack action but he can never get more than 2 attacks (with one potentially being a cantrip) because they do not stack.
Understand?
Now if you changed this and further homebrewed these rules such that different extra attacks all stack, then I agree it would be OP.