Because something like Ich will just keep re-infecting the fish over and over. If you don't catch it, you'll infect every fish that goes in after it, if you do catch it, now you have to deal with Ich in your DT. Assuming you don't want to put copper in your DT, you'll have to get a QT set up and the DT will have to run fallow for 76 days.
No, you can put a fish in an uncycled tank, but you'll have do a lot of water changes and stay on top of checking the parameters. It's a lot easier to change out 5 gallons in a 10 or 20 gallon tank every day than a similar amount of water in a larger DT. Also, medicating a larger tank requires more meds (IOW, it costs more).
Personally, I'd set up a QT the fish in a separate tank while the DT is cycling.
If for no other reason, you're going to need it anyway if you plan to QT fish, which I assume you are. So this way you'll have it all set up and ready to go. Plus, if you need to pull a fish out of the DT, you know you have everything on hand so the only thing that will slow you down is waiting for the water to heat up. It's better than trying to think of everything you'll need, hoping you don't get home and realize you forgot a heater or a power strip or something like that. If you plan to medicate (instead of observe) in the QT, you'll have meds on hand also.
Short answer, if you're going to QT everything, you should, IMO, QT the first fish going in as well. It doesn't seem logical, but it's the right way.
For what it's worth, I had the same question as you when I started stocking my tank. I reluctantly set up a QT knowing my first two fish wouldn't be in the tank for at least another month. I ran them though copper and all the meds Humble Fish recommended and they still managed to have Ich or Velvet or something (I don't recall what we decided it was). It ended up taking almost three months to get cleared up. And the sick fish looked 100% fine when I got it and all the way through copper. Whatever it was, I'm glad I didn't introduce it to the DT.