M.R.
Not familiar with this program but I do know a bit about Odyssey of the Mind. These are competitions, and that can be great, but be aware that they are competitive academic pursuits with coaching and the pressures that come with that--as Kelly notes below. A ton of fun, yes, but still pressure. (We did another academic competition for years in elementary and middle school, so I know what these are like -- fun teamwork but there is pressure involved even if it's the kids just pressuring themselves to do well..) Just be aware of what she's getting into. And you may hear that there are a lot of requirements, non-negotiable ones, to be at certain practice sessions etc. as competitions approach.
Girl Scouts is not comparable to these programs at all, so please don't think you're replacing one with the other. Totally different organizations, goals and activities, and expectations. So to me it's really not a matter of one "versus" the other except in terms of what your child would prefer.
By the way, wearing a GS leader hat here: Your child is NOT required to be in her "school troop." Though many troops are school-based, with the kids being all from the same school, that is done as convenience -- it is not some kind of GS rule though it's treated as the norm in a lot of places. If you want her to continue with Daisies, contact your Girl Scout Council by going to www.gsusa.org and searching for the Council that covers your part of Illinois. Contact them and say you want to be put in touch with the manager of the "Service Unit" for your specific area where you live. Ask the Service Unit manager to help you find a troop for her. Most will meet at schools after school but there is no real reason she "must" be in one at her school.
Finally -- not sure what you mean by the "school troop was already in place." Did they tell you that they were not accepting any more girls and the troop was full? That is the leaders' choice, as leaders can set the size limit for a troop, but if the issue wasn't a "full" troop in terms of numbers -- why wouldn't she be able to join at any time? You don't have to start with a troop at the very start of the school year. Our troop has always picked up girls throughout the school year!