CPG Grey, Which Planet is the Closest?: https://www.youtube.com/watch?v=SumDHcnCRuU
To be fair that arrangement is rather unstable, especially with planet-sized objects.
talk about cosmic apotheosis.
So, if you when you say "the sun" you mean the sun surface, then yes, the sun is always the mostest closest celestial body, to all planets
Furthermore, since the sun is also orbiting around the shared center of mass of the whole solar system, this displacement albeit very small, is still enough for me to not intuitively understand if it makes the sun's center of mass closer or farther away on average than the closest orbiting body to the sun
Since that gravitational center, and the center of the pairwise systems is not the same, I wonder if a planet at that place is really the best solution.
It's not half the time and it is half the time. If you include the Sun as well as one of the possible answers (which I'd argue you shouldn't because neighbour implies same significance, not higher), the answer would've been an even split between Mercury and the Sun (on a large enough time scale).
If Mercury's year somehow lasted longer than a year on another planet, only then would Sun be the clear winner.
Think of it this way: If we take the Earth as stationary and just look at the respective motions of the Sun and Mercury, then the Sun is also (roughly) stationary* and Mercury moves around and around it, sometimes close to us and sometimes far.
Now, if Mercury actually yo-yo'ed through the Sun, then you'd be right: exactly half the time it would be closer to us, and half the time it would be further from us.
But it doesn't yo-yo through the Sun, it moves in a circle. When it's 90º from us and the Sun, it's still further away from us than the Sun is. So it has to get even closer before it's equidistant. So it's actually closer to us only less than half of the time.
*Yes, the Sun would also appear to orbit around the gravitational center of mass, but this doesn't affect the thinking above.