You can still switch the same game between the TV and on the go, if you get a handheld version and a hybrid version (or a hypothetical home console only version). It's rather telling that Nintendo are doing cloud saves as part of their subscription service imo, since it's most useful in scenarios where you are switching save data between different devices.
I think some vague confusion around branding for an already successful and established platform is far less of a concern than literally millions of customers that are holding back from buying a Switch right now, as evidenced by the 3DS's healthy hardware sales even today.
I'd also say that getting people used to the idea of a "Switch Go" or whatever that was a portable only Switch machine is easier than selling a device not called Switch that plays Switch games.
It's just a brand, it can mean whatever the brand needs.
I'd argue a dock-less Switch would do a better job achieving the results Nintendo wants. Removing the dock, grip, and HDMI cable (maybe charger). would save a lot, and get the Switch to a comfortable $199 price point. Nintendo's already toying around with this in Japan.
It wouldn't save anywhere near as much as a full redesign focused entirely on cutting costs.