That's a big house... Probably not wise to cover it all with so many discrete mini split systems. Do you have funding readily available to consider a big investment in Geo?
The issue is when you way mini split, do you mean ductless or ducted?
In the HVAC industry, these terms can get confusing since most manufacturers offer a inverter compressor HVAC system.
Another variable....
If you have a 5k sq ft house, depending on the way it was built with insulation, it could require a total of over 60,000 BTU/h of heating and cooling, OR it could require less than 24,000 BTU/h of heating and cooling.
My own home is 3,200 sqaure feet, and due to the window areas along with basement below grade, takes over 80,000 BTU/h for heating and cooling, BUT, the below grade basement and large window spaces on the 2nd and 3rd floor actually "compete" against each other for cooling and heating for the entire house (using 3 systems).
When you go with a single stage compressor HVAC system, doing gas AC, you may need 20-40K of heating with the gas furnace, but pretty much no AC, but by the time you add in ductwork, it makes no sense not to have the AC option for whatever reason.
The bigger issue people forget about with loads for larger homes is air flow meeting those heating and cooling requirements for various spaces using only 1 system.
A house that is 5k sq ft that requires 2 ton or less heating and cooling, the issue becomes meeting the air flow requirements and finding equipment with the proper blower speeds to hit those air flow requirements for the ductwork.
I also could be wrong, but I believe the OP is moving/building a smaller house.
We only have 3k sqare feet, and now that my wife and I are empty nesters, our next move/build will be definitely smaller LOL