The Switch was the logical successor to the Wii and their declining handeld market due to the stiff new competition from iOS and Samsung. People like the Switch.
I think they had the idea for the Switch floating around by the Wii/DS generation. Maybe by 2008 when they realized that they were having dev-capacity issues and perhaps losing software sales as a result, but they kept with it because their hardware was so high that all their games they did release were selling incredibly, and they were getting fabulously wealthy over the next 4 years.
I dont think Wii U having a screen had much to do with influence on the Switch. The assymetrical gameplay and here you can look at a menu instead of wasting 0.125 of a second pressing a button to see it instead were gimmicks to justify the fact that they only had half-ass levels of the Switch tech they wanted. Releasing the Wii U because they felt they needed a new console at that time, and that a simple Wii HD would not be enough. The Wii U was clearly not in development for long, it was rushed out with a lot of problems (clunkiness and tremendous load time going into games and navigating the OS - issues the Wii didnt have).
I describe myself as a little dose of toxic masculinity.