Why does it seem quite unlikely? As you rightly point out, we don't know exactly when Papias writes. Hypothetically he could date as early as the 80s, really.

Papias' note that the presbyter John lived to his time of writing and thus to old age seems supported by the author of 2-3 John, who labels himself as ho presbyteros (just as the John who was alive at the time of Papias' writing was called) and seems to be the leader of the Johannine community (be gives commands to his audience and expects them to listen, etc) and applies the eyewitness language of John 19:35 and 21:24 to include or talk about himself in 3 John 12 ("We also testify for him, and you know that our testimony is true"), and so the author of these epistles was probably the Beloved Disciple (so Urban C. Von Wahlde's 2010 Eerdmans commentary on the Gospel of Letters of John, Vol 3) or an amanuensis (though I favor eyewitness authorship). This is supported even more directly by John 21:20-23, written around 100 C.E., which indicates that the Beloved Disciple lived to very old age.