PSvita is also a home console, when using same logic as with switch.
There is even home console version of vita, the vita tv.
Calling switch a (pure) home console is just false, yes it can be used as home console, but it is basically a handheld with tv out and there is handheld only model, so it is more of handheld than home console.
It is its own category and most fair thing to compare sales wise is to compare it to combined wii+ds numbers, as it replaced both kind of devices. Comparing it to home console only systems (ps5+xbox) is kind of not logical.
Home console should be a system, that can only be used with TV, anything else is either a portable, or something different. Nintendo also have kind of different audience than playstation/xbox, so it is not like many people choose between them three, they just want nintendo and nothing else.
I have ps5+series x + switch
Switch have never been sold out in my country
PS5 have been sold out since the launch
Series s have been available since the launch
Series x have been mostly sold out, but also available for some months in big quantities, like it have been available for weeks now
So basically same as gens before, playstation always sells out, others dont. But difference is that ps5 have been sold out for years instead of few months. And on world wide scale, PS5 have been the only truly supply constrained system of them all, as switch + xbox have never been sold out on global scale