In most countries (I'm open to counter-examples) a president is expected to be above the fray of party politics. If picking a sitting minister is a serious possibility, then it's clearly not a consideration in Germany

We would expect them to switch to presidential mode the moment they are elected. If they don't manage that, we could try to force them to resign. By the way, all our presidents with only one exception were catastrophes.

And I have no idea why the SPD or Grünen would contribute to electing a CDU partisan

Because they can't dream of a majority without Die Linke. And a coalition with Die Linke is impossible because all members of that party are dirty and stink, and are stalinists, and eat babies for breakfast. So, Greens OR SPD can form a coalition with the CDU but not with each other.