Yes, a hypothetical planet located at the centre of the sun would be every planet's closest neighbour, by virtue of never getting as far away as others.
Nit: located at the gravitational centre of the solar system. Which is not the perfect center of the sun (though still inside it) since all the planets pull on the sun too.
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.
> Is the Sun not our neighbor? And it's closer than Mercury, isn't it?
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 90ยบ 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.
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.