I do believe that the Greybeards explain to Dragonborn that only a Dragonborn can shout, right? They also say that he/she is probably the last Dragonborn. Well, then they seem to spend the rest of their time teaching him how to shout by doing it themselves? So then, if only a Dragonborn can shout then how come can they do it?

I think only the Dragonborn can shout innately, others can do it but takes them years of training.

Pretty much this. Only a Dragonborn can do so without special instruction and training. The Greybeards have spent years learning to do so, and while they could technically learn the words from a word wall, it would be like reading it from a book, you'd understand what it means, but it wouldn't give you the power as it does the Dragonborn.