Why would the game specify the disappearance in the spell description then?
Because it does NOT have to be absolutely simultaneous with the appearance. As mentioned, an "Instant" is an unspecified period of time.
So, once more, you can very well use
@Maxperson's perspective, and it works, or you can use other perspectives, and they also work perfectly within the RAW, even assuming that it's important to you. And that's my only point, the game is open, if you want teleportation to be more Star Trek like (because it looks cool), you absolutely can as well.
Same if you want teleportation to go through other dimensions in your cosmology (like astral or ethereal in previous editions), you can (just being aware that this introduces constraints on the spells, like it was in 1e where the absence of a medium prevented some spells from working in some places), but if you want it to be purely (intra)dimensional travel, you can too.
And for me that's the beauty of 5e, the rules are clear enough to run games, but they do not box you in, reducing your possibilities, but also leaving you open to ruleslawyers and nitpickers, since the DM can make any ruling he wants.