AFnord: You just pick up your packages when you do your grocery shopping.
And I have 4 different pickup points within convenient walking distance. If there is a manned gas station or store larger than a hole in the wall nearby, then you are bound to have a place where you can pick up your packages.
Also, Sweden is really sparsely populated, so we have much the same issue as the US. Also, I would pay extra for not having to be home at awkward times if I was given the option when shopping from US or UK stores (US ones tends to be worse in this regard, most budget shipping options from the UK will just deliver it to a pickup point) ;) Luckily free shipping is common here, and it is rare for shipping to be more than 30sek (~3€) for anything that is not really heavy.
No, you guys don't have the same issue we have here. There's still a fair number of people receiving their mail via horseback and helicopter in the US. Even within cities it can easily cost more than $5 to go retrieve a package. Not to mention taking an extra hour or more, if you're not somebody that drives. It's barely any better if you're already out picking up groceries, you still have to find a place to park which can be an incredible hassle. And the traffic, it took my Dad almost an hour to leave the city yesterday.
As far as sparse populations go, the numbers you're looking at aren't accurate. Alaska is pretty freaking huge and has a tiny population, but places like New England that are ridiculously over populated make up for it. It might appear that we're less sparsely populated, but those sorts of nationwide numbers really don't reflect just how unpopulated large parts of the US really are.
So yeah, it might work in Sweden, but I cannot imagine this working in the US, the logistics of it would be a nightmare. Not to mention the relatively small number of chains that are truly national, requiring a huge number of deals with different businesses to ensure reasonable coverage.
All in all, I just can't imagine it being worth the effort when they can drop it off on my doorstep for less than the cost of picking it up.