FDR or no FDR, that saying has always irked me. If we're generally strong people with a normal amount of courage (or even particularly strong people with a particular high degree of courage), we tend to fear only what is unthinkably horrific to us (and what would, in fact, be unthinkably horrific to experience). So, we fear what we fear for good reason - and we don't fear "fear itself" (at least those of us who fear the right kind of things).
There are times when, again for most normal human beings and particularly courageous ones, as well, we overcome even real reason to be afraid out of purpose, the drive and the need to protect someone else, the grown-up understanding that must overcome our fears, and often with the help of adrenaline.
That "line"/"saying" was said within a certain context, so I don't find fault with the words, themselves, in relation to the context and time in which they were said. My "issue" with those words is that they are often taken out of context and presented as "words to live by in general".