Yes. Backwards compatibility isn't ride or die, but it's really convenient.

Sony set the standard for home console backwards compatibility with the PS2. I know some other consoles had done it in the past, but they weren't as major (Atari 7800) or required an adapter (Sega Mega Drive/Genesis). And of course the Game Boy Color set the stage for portable consoles.

For the Eighth Gen, the Wii U supported all Wii games. The Xbox One introduced limited backwards compatibility with the Xbox and Xbox 360 from 2015 onward. And Sony famously has no backwards compatibility on the PS4. And the PS3 started with PS1 and PS2 BC, but then widdled down to PS1 only for most models.

If the Switch's successor (which I would consider Tenth Gen) uses cartridges again (and I bet it will), it really should be backwards compatible.

Now that a majority of my game purchases are done digitally, I 'd say (for me at least) yeah, BC needs to be there day one. If these companies are serious about cultivating loyal customer bases and pulling in new consumers, they must support all of their digital purchases past and present going forward from this gen on. While I plan on keeping two of the three PS4s in my home and upgrading one at launch I don't want my family or our library of games to become fragmented. And I most certainly won't be upgrading if that happens.
I also think the value proposition is not there if they cannot support past game purchases. People will be less likely to invest in their ecosystems if they have to plunk down full price for a game to only lose access to it. At the very least it will create more agnostic consumers who can take their dollars to the next company with a better value proposition. Not saying that's necessarily a bad thing for us as consumers, but for these companies to keep a level of customer loyalty I think it's a must that these next consoles support BC. Besides, we all know the architecture is going to be largely the same so we know it should be possible to do right out of the gate and without emulation.

I think there will be if not 100% complete, widespread BW compatibility... But I don't the OP's premise holds up,
that this encourages people to buy early... which would logically mean increased sales, yet AFAIK platforms without
BW compatibility have had no problem reaching record-breaking sales, I just don't think that is real factor in sales.

I honestly don't care about BC. I don't have the time to play all of the games that i want from this gen, so i rarely will have time to go back and revisit games from previous systems.

I don't sell old consoles so I don't really care about BC at all. Have a PSX, 2 PS2s, 2 PS3s, 2 PS4s. Same for an OG XBox and a XBONE, and a few others. BC was a bit of a deal for me back when the PS3 came out but that's gone and done. If a console has BC it's a nice goody, otherwise I don't care for it. I collect.

The rumor mill is suggesting that the PS5 will be compatible back through the PS1. It's a selling point,but for most the feature would be underused. Those that really want to play PS1/2/3 games right now will find that the consoles are cheap and sharp video solutions readily available:

Where? And whats a rumor mill? I cant find anything atm.

Sony already confirmed (twice) PS5 will be bc to PS4. Thats it. Why would they hold back information when the main part of the BC feature has already been revealed?

I just wished that backwards compatibility would be a standard feature, and not something that needed speculation.

BC is a huge seller for me. I only have so much room for consoles. Not only does the hardware take up space, but that is an HDMI input and a power input. Add in extra controllers... yeah, being BC is a massive get, especially if the controllers are BC.

Essentially being able to use my ps4 controller and upgraded HDD, makes the ps5 $160 cheaper...  does that matter?  Absolutely.  

And for the sake of argument, perhaps I can sell my ps4 for $150...  add that to the $160 savings, and now I am saving $310.  In my book, that is a massive selling point.  

Biggest seller for me by far. I always highly value being able to play any game from any year I want on my PC, just as much as I would on console.