This is how LCOs work. Each housekeeper has a block of rooms. If you have someone who wants a late checkout, now that room isn't able to get cleaned until a later time. Too many of these rooms, and the housekeeper cannot get them all cleaned by 3:00 -- check in time. So what Disney does, is they allow only a certain amount of LCOs per housekeeping block.
So, it's very well possible that you can be denied a LCO, while someone else requests one an hour after you did, and gets it granted.
LCOs are always discouraged, as they prevent guests from checking into a clean room in a timely manner. Although checkin may be at 3:00, guests get pretty upset when they can't get a ready room at noon.
As far as I know, there are changing rooms and showers available at all resort pools. Luggage can be stored with Bell Services or Luggage Assistance. These options usually help greatly reduce LCOs.