Can I use the Wizardly Quill to make a free copy of my actual Awakened Spellbook?

I know the Order of Scribes can already make a copy of the Awakened Spellbook by making the ritual in another blank book, but that leaves the old Awakened Spellbook in blank. I want to have both Spellbooks in hand for pure Roleplay.

Here’s the rules for Wizardly Quill from Tasha’s Cauldron of Everything:

As a bonus action, you can magically create a Tiny quill in your free hand. The magic quill has the following properties:

  • The quill doesn’t require ink. When you write with it, it produces ink in a color of your choice on the writing surface.

  • The time you must spend to copy a spell into your spell book equals 2 minutes per spell level if you use the quill for the transcription.

  • You can erase anything you write with the quill if you wave the feather over the text as a bonus action, provided the text is within 5 feet of you. This quill disappears if you create another one or if you die

Here’s the rules about replacing spellbooks from Player’s Handbook:

Replacing the Book. You can copy a spell from your own spellbook into another book— for example, if you want to make a backup copy of your spellbook. This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell. You need spend only 1 hour and 10 gp for each level of the copied spell.

If I’m not wrong, I don’t need to spend nothing to test the spells I already know. Just need the right amount of ink and time to copy them. The Wizardly Quill is exactly made for this purpose of saving time and ink. That’s my interpretation. Am I wrong?