I could see a busy service desk agent putting the new flight into the reservation but not reissuing the ticket.
OP - were you confirmed in First on the original flight? If not, maybe either the system dropped the seat because the eticket didn't match the reservation, or someone else wanted that seat and somehow somewhere it was taken from you for another confirned reservation with the ticket properly issued.
Either way, I always make sure inside the particular trip, after routing or cabin changes are made, that the ticket shows the current date as the date of purchase. You can see this when you're inside the reservation - online, top right, view receipt, or in the app, bottom left, view receipt.
It IS weird that you had an eboarding pass, though, for the new flight. I could see a paper boarding pass issued with the eticket not properly issued - but never heard of that happening with an eboarding pass.
OP - if this ever happens again, you need to speak to the Ticket Reissue Desk. The GA can call the reissue desk with priority to get the ticket fixed during boarding. Why this GA didn't is beyond me, but is obviously unprofessional behavior.
While I have heard of people being offered 10-15K miles for broken IFE, I've never heard of anyone being offered $250 for such (maybe long D1 flight). I think $250 is fair, but OP, if you don't, I think max they would offer is $400 for a Silver with this issue. I would just advise you have a goal in mind. DL will obviously review the situation with the employee and discipline as appropriate - but you won't know the outcome here. I believe they have apologized, so the only other thing they can do is offer compensation to try to recover.