Not so. Selim was grimly Sunni, but Bayezid II favored the Kizilbash (Twelvers) and purging them was a big part of Selim's start in power.
But apparently he wasn't a Twelver himself, or, at least, I don't know any reliable source suggesting that.
Also, the Kizilbash were not very orthodox by Twelver standards either prior to Caldiran, and in part even after. They would get purged by the Safavids as well.
I agree that the crystallization of religious options by the Ottomans and the Safavis was for a good part more about politics, power, and contingent expediency than actual belief.
The Safavids moved to a rather fringe extreme Shii ideology to the more established and moderate Twelver outlook in a matter of a few years, and it would not be unconceivable to have them endorsing Sunnism at some point, though I don't think it's very likely.
On the other hand, the Ottomans had used to welcome or at least tolerate a hell of a lot of Shii heterodoxy in their early times and weren't big on Sunnism at their beginning.
I still think that more or less any time after the fall of Constaninople would be to late to endorse officially a form of Shiism as the official dynastic ideology without a spectacular backfire, especially if they try top-down proselytism.
The Safavids have been fairly unique among Shiite dynasties I know of for even attempting that, not to mention having that succeed and stick. I suppose that some parts of Yemen might offer very broad paralleles, but I don't know much of that area.
Usually, the Twelver or Ismaili dynasties before them did not seriously to spread their heresy over the subjects provided theey didn't rebel. On average it was more of a Sunni thing I'd say ( the Almohads are an example, though even there, it was hardly a lasting success). I think the Ottomans, in the unlikely case they adopt Shiism, would leave the Sunni masses largely alone as the Buyids and the Fatimids had usually done before, or the Shii rulers of Awadh would do later on.
(Yes, this seems to contradict what I stated about the Fatimids being consistently missionary. They were both, engaged in missionary activity and basically tolerant of a majority under their rule that professed the religion of their enemies. Stranger things happen.)