Faxon was, and probably still is, a truly exceptional putter; in his prime, he was generally considered to be one of the best ever.
Which probably means that, among other things, he has exceptional eyesight, including peripheral vision and depth perception. In tournament play, I play with very few older guys who want the stick out for putts of any length; at the most, some guys want it out for shortish putts, and perhaps the majority now leave it in for ALL putts.
All the studies that have been done about leaving the stick in vs taking it out are about what happens if a putt hits the stick, which means hitting the hole also. And since most putts of any length do NOT hit the hole, and therefore the not the stick, what the studies don’t measure is the proximity of a 30’ putt with the stick in vs out.
I’m almost 72, and my right eye is damaged to the point that I’m effectively playing with one eye, which means VERY poor depth perception. The flag in helps me; it’s a bigger and clearer target on putts of length.
As to the “bad look” thing, compared to what? Somebody tending the flag?