You, and they, are wrong.
The Constitution requires a "System of public education" - not "A system of public schools". Once the vouchers are defined as part of the "system of education" - they are by definition part of the "system of public education".
The intention of the wording in the Constitution is that the state is responsible for a system to educate the public. Not that the state is responsible for a system of public schools.
The word "public" refers to the people, not the school.
I went back and re-read the introduction to the decision, and also read through to what appears to be a more complete text of that article on page 15. The parts of the Florida Constitution that they reference there does not use the phrase "system of public education". It does use the phrase "free public schools". Can you point out to me where the phrase "system of public education" appears?
The wording that I see in the introduction and on page 15 is:
It is, therefore, a paramount duty of the state to make adequate provision for the education of all children residing within it's borders. Adequate provision shall be made by law for a uniform, efficient, safe, secure and high quality system of free public schools ....
It seems to me that this wording quite explictly does mean that the State of Florida is responsible for a system of public schools, and sets forth requirements for them as well. How do you interpret the above wording as not requiring the State of Florida to maintain a public school system?
Note, too, that this puts the State of Florida on the hook to educate any child resident in Florida, regardless of the legality of their presence.