I'd say that Nintendo has some TV experience due to Captain N and The Super Mario Bros. Show. Then there's all the anime they've had done like Pokémon, Kirby and F-Zero. I'm not sure that would have been much of an issue and frankly what has Harry Potter done in TV anyways? It's a film franchise.
So, in that regards, Nintendo doesn't have a great resume. There's the Pokémon movies and Super Mario Bros. and The Wizard. Can't think of anything else beyond that. Even then, I'm not sure how much work Nintendo actually did on those projects.
As far as the videogame side goes, I'd have to start looking up information on the series but does anyone know how well those games sold? How do the sales compare with Nintendo software sales? It's easy to say that making it 3rd party means more sales but if the first couple games suck quality wise, they could ruin the potential sales for the rest of the series and may not make that much money on multiple platforms compared to a single platform but with a high quality level of game design.
It's funny. When I think of the games, I just think cheap movie tie-in / cash grab but I guess the games really could have been based more off the books. I don't know since I have no expertise on anything Harry Potter related. Still, to my knowledge, I think the games were released around the same time as the movies which is probably why I thought that way. Not to mention I've forgotten they've even existed. It's not like anyone ever listed a Harry Potter game as one of the best games of the year for anything.
I imagine Nintendo's approach would have been to release the games for it first and further build up demand and then do things like a movie or TV show for the series. Similar to Pokémon's approach or their other videogame series really. I suppose it is smart in that it doesn't look like the game is being based off other media which could hurt its perceived opinion of it like other game adaptations.
In the end, I'd say it really comes down to the movie rights. Perhaps if Nintendo had only pursued videogame rights, maybe a deal could have been worked out but I'd say the most profit made on the series besides the books was the movies and the games aspect of it was just an extra bonus for Rowling. No doubt, the lawyers or people on Rowling's side recognized that and knew that Nintendo would not be able to handle that aspect well and was most likely focused more on the videogame side of things which would result in pursuing a less financially advantageous situation for Rowling and the brand.