No. Nintendo would go mad with it I think. There are already so many Pokémon games being made. I don't want even more people at Nintendo making them.

Game Freak might not want to be bought

If the price is right.

Nintendo would get very little benefit if they did.

I was trying to sort through the legalities of who owns what rights to Pokemon, and all I could tell is that it's pretty complicated. But the short answer is that Pokemon can only be released with Nintendo's blessing and that is the real thing they need. Nintendo basically controls the Pokemon IP, even if Game Freak gets most of the direct profit from it.

That's a misconception that mobile and 3DS compete the actual truth is they're different branches as the games people go for on each are heavily different, mobile is essentially another platform for Nintendo now. It's like saying Facebook games compete with Steam.

Nintendo actually owns not only controlling stake but also full publishing rights of Pokemon, they fully own Kirby as well.

Nintendo is the main owner but HAL have some stake and manage the IP for example the Kirby animated series was developed by a studio that required Nintendo's collaboration even though it was solely distributed by HAL. Essentially Kirby is handled in s similar way to Pokemon but not on the same scale, this frees up Nintendo to focus on other things.

They own the controlling stake because they own significant stock in Creatures Inc. this gives them controlling pull over the franchise they also own the trademark fully as mentioned on the Pokemon official site.

Nintendo already secured Pokémon rights when they bought 33% of the Pokémon Compagny

This. And this is the reason why I'm pretty happy with the current arrangement. Pokemon is bound to Nintendo, but it is also bound to GameFreak as creator and their creative freedom. This is a good thing, and it should stay that way.

They do list them though but as an affiliate like Pokemon Company.

Creatures are the former developer known as Ape Inc who created the Mother series, Nintendo created and bought significant stock in them the same way they did Project Sora, the are many companies that are subsidiaries but operate with their own identity and stock as one of many such examples of different business handling of subsidiaries. Given how the stock information on Creatures is kept under wraps and how they operate the are tell tell signs Nintendo own the controlling stake through certain business means where the details don't require disclosure as owning even 1% means partly owned but even that isn't disclosed.

Creatures have also never been referred to as Creatures when they did appear on financial papers as even after formation they were still called Ape Inc on papers as seen in this document here from 2017 which lists them as an affiliate that doesn't adhere to certain circumstances. The document highlights Creatures as having no impact on other financial aspects that concern the report so is exempt from being talked about. Ape Inc. is likely the actual company name for Creatures with the latter being a trading name.