Are passengers expected to check in based on their original time, and suffer in the airport as opposed to following a flight delay officially communicated via EK?
Unfortunately, yes I think they are.
I'm going along with Emirates' approach even though I wouldn't like it happening to me. The contracted check in staff could have other duties when your original scheduled check in period finishes so they would be gone. I think the text message probably was a wrong move even if it was sent with the best of intentions as it led you to asssume check in was delayed when actually it was only the flight.
I had a similar situation a couple of months ago with QR when I knew my flight was being delayed and chose to go 24 hours later rather than turn up at the airport and sit there for 6 hours or so. Mine was exacerbated as it was leading to a missed connection, and a further 6 or 7 hours in DOH, and this probably made them more amenable to change the entire itinerary as it was going to be changed in some manner by the missed connection. If it was just the one leg I'm not sure a change would have been possible.