At EDI, BE and AF/KL check in next door to one another, yet the staff have nothing to do with one another (well they might go on tea break together). The BE desks as well as the ticket counter have a sign which states that for the flights to CDG you have to check in at the AF/KL desk.
As AF/KL and BE use totally different check in systems, staff and handling agents then you cannot simply walk up and check in with BE if you're flying on KL/AF. You can, by all means, buy a ticket for an AF flight on the BE website, but then codeshares are commonplace all over the world.
For example I used to work for a star alliance carrier. If a person who booked a flight on another Star Alliance carriers website came up we couldn't even see their booking let alone check them in, even if it was a codeshare with the airline I worked with.
The check in machines are useful if you have hand baggage only. If you have bags to drop off they are essentially useless unless you want to get a specific seat and then wait in line to drop off a bag. Often though it is cheaper for airlines to have a machine print out the boarding cards then to have a person on the desk do the same thing, so these machines are usually promoted. I imagine you could have simply gone to another machine and requested a boarding card reprint, not sure why the staff didn't give you that option. Certainly when I check in using a mobile I always choose to pick up my BC at the self service machines and have never had an issue, so they can surely simply reprint a card.
As someone else said, lines for the likes of AF are long and will take a lot of time. Passengers travelling to India or Africa with 52KG of hand baggage aside, you have to also take in to account the stringent visa and document checks which have to take place for some of the more exotic destinations and the fact that someone at the front of the line will be kicking off about being sat apart from their beloved on a 1.5 hour flight.
One day, if every airline uses the same check in system you might be able to dump your bag wherever you like, but until that happens then only certain people can put those 3 lettered stickers on them, I'm afraid.
