Mini splits have their advantages, but they just don't make economical sense in a home (or business) with 10-15 rooms because you would need 3 or 4 separate systems to keep each space conditioned effectively. So your installation cost would be 2 to 3 times as high, and if all units were operating regularly your heating and cooling cost would be greater. The reason you see more of them in European countries and larger cities in the US is because of the way their spaces are set up. If you have a small one or two bedroom apartment, a mini split with maybe one duct from it to the bathroom is a great way to go, but keep in mind it is recirculating its own air with no way to get fresh air in most cases.
I am installing a mini split in my bonus room since it's only one room that will be a much different environment than downstairs (heat rises, doors being open/shut downstairs, etc.), but I am installing a fresh air central unit in my home due to using foam insulation. A central unit in a home with traditional insulation types has the opportunity to get some positive fresh air return from the attic, but in my case the roof is sealed air tight with foam so the unit needs a source of fresh air which I bring in with secondary plumbing from a location in the soffit. This type of system has a ton of efficiency benefits, but also gives the home the fresh air that it needs to prevent the environment from becoming stagnant like in cases that this thread is addressing.
I'm a big fan of mini splits and try to incorporate them as much as I can, but more times than not a central system is what gets installed.