If one reviews the Amtrak site and the British rail site, I think it's obvious that there is no easy or accurate count comparisons between the many Amtrak stations nationally before commuters, and what appears on the British rail's map to be about 22 non local British stations not including commuters. So I'm not going to waste time chasing my tail to try and prove what I feel is more than obvious to anyone who spends a little time on both website! Believe what you will; I'm moving on!
What the hell are you talking? Britain has 2,000 railway stations, of which I'd estimate very roughly (so while the total figure is definite, the divisions are very rough guesses, so don't start analysing these numbers), about 200 receive intercity service, 800 receive medium distance regional service, and the remaining 1,000 receive what is equivalent to America's commuter services. The national rail timetable, between me and the stationmaster at the station in my hometown jokingly known as "the Bible", is a 3,000 page document.
So the suggestion of 22 non-local stations is one of the most hilarious things I've read all week, even if it is only Tuesday.
From my experience, the UK system will let you choose to pick up tickets from a machine that isn't the start of your journey - but it does point this out to you. I've heard reports that one can collect tickets from any machine (not just the one specified), but I've not tried this. I've also picked up tickets 'over the counter' from a station. Normally I get them mailed to me - often the advanced booked UK tickets aren't refundable anyway, so theres no point in leaving collection to the last minute over here.
As to how useful it is to be able to pick up tickets from another station - one case I can think of would be of use would be a commuter who could use the machine (or counter) at the station the work near (which is likely to be in the city) to pick up tickets for travel from their local station, which maybe is unstaffed.
It gives you big red letters if it's a different station, which certainly helps. I think it does have to be the specific station, and one would have thought it very easy for the recognition of your credit card to take you to, effectively, an account in which all your purchases are viewable, so that you can print any out anywhere.
And yes, the way you suggest about using the machine at the station one commutes into is exactly what I do; I visit Leeds City every day, and when I need to, I pick up tickets there to travel from unstaffed Burley Park, or Appleby, which is staffed but without ticket collection.
The Downeaster has a fantastically simple solution, driven by the necessity to offer advance booking to all passengers as the service requires reservations (I've never known it be full, but I suppose at very popular times it can happen). The conductor has the tickets of the passengers boarding at unstaffed stations that have booked online or over the phone, and I think identification is required to present to the conductor in order to prove the tickets are yours.