An alternate idea would be to go in the opposite direction and separate the home console from the handheld again, but have them sharing functionally the same library. Nintendo's own statistics show that the vast majority of their users only use it as either a tv console or a handheld, with few using it as a hybrid. The holy grail that Nintendo could never obtain was unifying their handheld and home console libraries, because the technology just wasn't there. The technology is here now, and they can do that without the hybrid setup, which has its own flaws. The hybrid design's main issue is that its a jack of all trades and a master of none. For a handheld, its got a big, bulky, and not very ergonomically done design that even in your hands feels more delicate and breakable than the denser, more compact, and more stout handhelds of old, and it can no longer be comfortably carried around in, say, your pocket. Also, it has a terrible battery life for a handheld, with the only fixes being things that weaken its pickup and carry nature, like a battery pack. As a home console, it underperforms its competitors by a noticeable margin, and even when connected to the TV, the console's capabilities are limited (even BOTW stuttered in places, and this feels like the first console that actually limited Nintendo themselves due to its technical limitations). Because it must also be a handheld, there is little Nintendo can do to up the power, in that form factor, that won't also jack up the price.
Separating the two back into two separate machines would fix these issues while allowing the same cross-format gaming. The handheld could be designed from the ground up as a dedicated handheld again, with a better form factor (more in line with the Switch mini or even smaller if Nintendo wanted) and a stronger battery life. The home console can be more powerful and more in line with modern capabilities (such as being able to use external hard drives like the Wii U), and not be limited by technical limitations. The two consoles would share the exact same library, and Nintendo could do this one of two ways. First, Nintendo could have both consoles use the same cartridges, thus making them interchangeable. If you put the cart into one console or the other, the game would detect the console type automatically and adjust the game's settings automatically to fit whatever console it was in, emulating the "docked" mode and "handheld" mode that the Switch currently uses. Or, second, Nintendo could have two different media for the two; the home console (which I would tentatively call Nintendo Switch 2 Home) would use a CD based medium (maybe blu-ray, maybe another third party proprietary disc like HVD, or a Nintendo created disc system), while the handheld (which I would call the Nintendo Switch 2 Mobile) uses cartridges. This second option has two benefits: first, it would allow Nintendo to formally separate the libraries of the two consoles, so that, while they would be mostly identical and have the same games, they could each get games that were unique to one or the other should the game developer choose to focus on one over the other (such as Rockstar Games choosing to prioritize giving Home a version of GTA, while ignoring the handheld due to hardware limits), and second, lets face it, optical media is the superior storage medium due to how much more data it holds, so there is no need to hamstring the home console by tying it to the inferior media.
To encourage people to buy both consoles, Nintendo could have them interact in some way. For instance, to take your wireless TV streaming idea, I'd suggest that the home console would be able to stream wirelessly to the handheld, and vice versa. This would allow, for instance, Wii U style off tv play, for the handheld to be used as a controller, and for the handheld and tv to be used at the same time, just like the Wii U, but with none of the Wii U's downsides, because its actually two consoles, not one. You could also setup a situation where the two consoles could be physically connected, like the Switch and its dock, allowing them to share processing power and unlock even better gameplay, like full 4K, but this would require a high capacity connection, like a Thunderbolt connection, but its theoretically possible with today's tech. Other than that, both consoles would share everything between software wise: apps, games, information, profile data, etc. can be accessed on one or the other via one's account, though individual games would need to installed or updated on each console separately. The Switch's games could be backwards compatible for whichever console gets the cartridges.
