Are you people serious... It's the First Day Badge. Not the first week, first three days, first 48 hours. First day = 24 hours. What's the point of having a "First Day Badge" if it's handed out within the first 72 hours?
Excuses: "Oh but some people have a life, Disney, and they can't get the badge."
You know what that's called? Tough luck. If I have work the day MMK releases and I miss out on the First Day badge, then tough freaking luck for me. To some people, including me, the First Day badge holds a level of prestige. --You were there on day one-- Don't lessen/degrade it by handing it out for the first 72 hours. Just the name alone loses all value because of the fact it'll be obtainable for the first 72 hours.
If 72 hours passes, change the name of it. The name of the badge will, quite literally, no longer mean what it says.
Monorail foresaw this and had a great idea. Since people can't cope with the idea that they'll miss out on something and they have to have everything be "fair." Change the name from "First Day Badge" to "Here Since the Beginning," or something along those lines. That would make more sense rather than handing out the "First Day Badge" during the first 72 hours.
EDIT: As for lag, that's the only foreseeable force that can stop anyone from getting the badge in 24 hours. Even if there were server issues that caused everyone to disconnect. You have T W E N T Y - F O U R hours. That is more than enough time for Amy to tweak things and allow everyone to get online, it was far less than 24 hours for Amy to fix the connection issues the first time on OVMK.