A "famous" photo has at least one element that holds attention over time - significance to history, aesthetic, an idea, and so on. Some photos have more than one attention-grabbing element.

Going out and copying a photo has its benefits, but the exercise for the sake of the exercise means the photo is more likely to not stand the test of time - the original photo might holds an element that may not be so easy to copy. Attempting to copy a "good photo" by replicating only its aesthetics will improve your technique; but you should be aware that the copied image might never be much more than a shallow, redundant imitation. Roger Fenton's Valley of the Shadow of Death or Edward Steichen's Pond Moonrise are important images, but the element that grabs attention is difficult to replicate as it is not aesthetic.

You might like Mark Steinmetz - many of his recent photobooks are collections of work made almost 30 years ago in the 80s and 90s, yet were only released less than a decade ago. A lot of these pictures were selected as he sat on them for many years, to see which ones continued to hold his interest.

You also mentioned posting right away, and it made me think of Steinmetz's quote in an interview: "Instant gratification as a method of operation typically doesn't yield work that will span the test of time."

Nostalgia is a strong factor for why a photo holds intrigue - especially for younger photographers (including myself at a time) who know/knew little about world history/photography history. Other elements are poorly understood, so the aesthetic is the only thing that can be read by anyone with a pair of eyes and not a lot of awareness of the context.