Well TEFAQ, as the name says, is specialized for Quebec migration.
The selling point of TEFAQ is that Quebec immigration only ask you to take the oral comprehension and expression. So in theory is easy to get a good enough score for your CSQ.
In contrast, for TEF ( or any other exam for that matter) you will need to take the 4 components and on the same sitting for it to be valid for immigration purposes.
The TEFAQ is directed to people that Quebec is their only option or plan. So why would they need to “waste” resources in other exams. The issue is that people don’t inform themselves correctly.
If you want some consolation, you will be able to use TEFAQ (if needed) for your citizenship application
https://www.cic.gc.ca/english/helpcentre/answer.asp?qnum=572&top=5