My family and I have ever stayed at the MCO Hyatt as well. Like others, we did save money staying there versus staying at the Disney Resort. Then we just woke up the next morning and went straight to the Magical Express Bus. Our room was not ready though. Grand Floridian check-in time is at 3pm. It's not a big deal though because you can leave luggage/items with the Bell Hop.
On other occasions we have also checked-in at night and just paid the money to stay the night at our Disney hotel. I found that when we checked into the Grand Floridian at night, there was not a lot of staff members to help us. We had to wait about 45 minutes because there were four other families in front of us. Our room was ready though. (This could have been avoided if we used the online check-in process. Maybe I'm just old fashioned, but I like the Cast Member interaction.)
If you're traveling with adults, staying directly at the airport would probably be easy. But I have to agree with
GADisneyDad14, staying at the Hyatt did not go as planned and "trip adrenaline" was very high.