Photons from a star entering earth's atmosphere can scatter off atoms in the atmosphere and be redirected, although it is even more likely that they will be refracted slightly by variations in the the density of air - this refraction by turbulence is what makes stars twinkle and blurs the images we see through ground based telescopes.
The fraction of photons that are actually absorbed and re-emitted is low, except at particular wavelengths of absorption bands - the opacity of the atmosphere is high at these wavelengths (like absorption by ozone, molecular oxygen, CO2, and so on). When a photon is absorbed and re-emitted, it usually comes out isotropically, meaning it doesn't have a memory of what direction it came from. When you look up and see a star, you are seeing photons that actually came from the star, after glancing scatters off turbulence in the Earth's atmosphere.
Photons of identical state are indistinguishable anyway, so trying to ask the question of whether a photon is "the same" after it underwent an interaction like scattering is a bit fruitless.
Betelgeuse is cool, but I feel compelled to nitpick that astronomers photograph supernovae in other galaxies all the time. The last naked eye visible supernova was SN 1987A in the Large Magellanic Cloud, a satellite galaxy of the Milky Way. But it was only naked eye visible at dark sites in the Southern Hemisphere, so many of us didn't get a chance. If/when Betelgeuse explodes (and as the OP says, it could be some thousands of years from now), it will be the first SN visible in our own galaxy known since 1604, and it will be hard to miss.