First, is this statement to suggest that handheld-only devices are no longer a viable object to sell in North America due to dangerously decreased demand?
I actually wouldn't say they are no longer viable. I would say that there currently isn't a significant market for them. (And heck, maybe the Switch is just meeting all these needs and we're only talking technicalities on definitions, and a huge chunk of % spend on Switch is actually for dedicated portable gaming and maybe we're both just wasting our breath and brains. Who knows.)
The right device with the right content sold at the right price could reinvigorate that market.
However can the "right" device be made, and what would that look like? Dunno.
Would the "right" content be identified, funded, and developed? Dunno.
Could all that be done at the right price? Again, dunno.
Second, are handhelds more often than not bought only as a companion device to a home console, or do they often get purchased as the only gaming device for a household?
There is ownership overlap, certainly. And it really depends on the age group. Portables generally lean younger, consoles a bit older. And you can see multi-portable ownership per household, something that's much more rare for consoles.
So are they bought primarily as a companion device? I don't think so, but the data doesn't allow for that kind of extrapolation, and I haven't done or seen any primary research (interviews or focus groups) that gets into those motivators. There are other companies that do that type of research, however.
And lastly, regarding the overlap of gamers who own handhelds and consoles, how much of that is brand-specific? For example, it's previously been implied that that there is a huge overlap of Nintendo console owners also being Nintendo handheld owners, and I'm curious how true that actually is, both historically and currently.
Well it's a bit more complicated now that 3DS is the vast majority of the portable market. It's hard to make comparisons as there aren't also MS and Sony portables of significant market size to compare to.
3DS and Nintendo console cross-ownership is at a higher rate from what I've looked at, but I don't know if it should be defined as "huge" overlap?