Will your 360 suddenly mysteriously die when you buy a new console? One thing both MS and Sony found out this generation is that the money spent on BC could have done a lot more making the console better. BC is only useful for the first year of a console. There is no hue and cry about the complete lack of BC in the PS3 Slim, if it's such an important feature, why not?
You've asked a number of questions there and I'll give answers from my perspective (which is all anyone can really do) as to why I think BC is a pretty important aspect in the next generation of consoles.
FIrstly, it's unlikely that my 360 will die when I pick up a Nextbox. However, it's even more unlikely that I'll want both consoles sitting in the limited space under my TV. I know some people don't mind having a jumble of electronic boxes and miles of cables sitting in that space, but I'm not one of them. One of the reasons that I haven't bothered to replace by dead PS3 Fat with a Slim is that I like the neatness that
just having a BluRay Home Cinema System, Satellite Box and a 360 sitting under my 50" Plasma. To ensure that I don't have a mass of remotes messing up the place, I have everything running from a Logitech Harmony (yet another reason I haven't been quick to replace the PS3). And why I bought into Kinect when I haven't so much into Move (or the Wii+Nunchuck).... less to clutter up my coffee table and media area.
So a Nextbox arrives to the market without BC, I'm not going to be making the jump anytime soon. I'll create a mess and that would annoy me.
You also make the argument that BC is only useful in the first year of a new console. There is probably some truth in that, but I think it would be fairer to say "in the new users first year", which could be 2 or 3 years into the new consoles lifespan. If I've invested £1000's in (for example) a 360, I still have many of those titles and enjoy them occassionally to this day. In fact many games I've bought over the last 12 months or so I still haven't finished, such as Dead Space 2, LA Noire and Comic Jumper. I fully intend to finish them all though when I have the time.
And the latter title I listed there is another key reason why BC is so important. When purchasing digital content, there should be a reasonable expectation that said content will still run on the next generation in the console chain. Without that, there can be no trust of buying digitally going forward. I bought Crysis 2 on Steam on release and I fully expect to be able to play that even when I upgrade the graphics card or processor or even operating system. I still have the original Tomb Raider 1 on CD and (with a little playing around) I can play that on my 2011 Windows 7 PC. A digital purchase MUST offer me a similar level of future protection, and where consoles are concerned then it has to be at least one console ahead.
The cost issue is a red herring. It's not about the cost of implementing BC in future consoles, it's about building Forward Compatibility into the console's hardware and software. That cost should be negligable with proper planning and a coherent vision.
And there was hue and cry when the EU release of the PS3 had 2nd rate BC, there was further crying when that was latterly built into the NA PS3's and yet more upset when it was removed altogether in the Slim.
So from my perspective, and maybe I'm a minority in the console consumer space, BC will be a key factor in whether I make the jump to Nextbox or not, and even if I do it'll certainly affect how soon (or not) that jump is made.