Classroom has no financial side
There is no price per student, no invoice, no PIX, no payment record and no overdue tracking. It was not built for that, so it is not a flaw. In practice, though, teachers who use Classroom still close the month in a spreadsheet.
A private lesson does not work like a class
Classroom starts from a class with a shared calendar. In private lessons, each student has their own time, and one-off lessons and last-minute cancellations change the amount for the month. The Aulency schedule works that way, and each student’s invoice takes those changes into account.
The parent needs to follow along too
In Classroom with Google Workspace, invited guardians can receive email summaries. Education Standard and Plus also provide links to preview Classwork. Those features follow schoolwork; in Aulency, parents can view lessons, reports and invoices for their dependents from one account.