Did Augustine teach sola fide? It is popularly taught that the reformers declared five distinctive elements of theology that differentiated them from Rome: salvation by grace alone, through faith alone, in Christ alone, for the glory of God alone, as sufficiently taught in Scripture alone. They quoted the church fathers in support of their views to demonstrate that they weren't teaching novelties or heresies and they showed particular deference to Augustine. But did Augustine actually teach the doctrine described in the five solas?
No.

Well, at least this appears to be Martin Luther's answer


  Augustine has sometimes erred and is not to be trusted. Although good
  and holy, he was yet lacking in the true faith, as well as the other
  fathers...But when the door was opended for me in Paul, so that I
  understood what justification by faith is, it was all over with
  Augustine. (Luther's Works 54, 49)
  
  It was Augustine's view that the law...if the Holy Spirit assists, the
  works of the law do justify...I reply by saying "No". (Luther's Works
  54, 10)


As to your statement about the popular belief that the reformers (and their descendants) frequently quoted a plethora of Church Fathers to demonstrate the legitimacy of their doctrine...

This is simply not true. Augustine is for the most part the only Church Father cited by Luther and Calvin.  The early reformers relied virtually entirely upon Scripture to formulate doctrine.

Reference: Calvin