Whilst I sympathize (and this is going to be hard not to come across as a dick...) there is sadly a lot of benefit fraud and without attending appointments and check-in-days it's hard to determine if someone genuinely needs it. If they make an exception for them they would also have to do it for the next person who could be wrongly claiming disability. Sadly not uncommon based on chats with a friend who worked there - Equal to the scum who scam blue badges off councils so they can park for free.

Maybe different centres differ greatly, but when I couldn't make an appointment when I was out of work they were fine with re-scheduling til the next day - providing I didn't ring up just before or after the meeting.