I cannot say why for certain the website makes these statements, but I believe it’s the problem of having to explain legacy parks and that the systems are not truly merged, and it will work at these parks, but not these parks. It is easier to just give warnings even if they are complete garbage.
But I can answer your question. I purchased a new 2026 Kings Island pass, activated it as part of the purchase by uploading a picture. Then drove to Cedar Point and used it there, before it was used at Kings Island. The pass worked perfectly, as did the add-ons. So yes my new pass from a legacy Cedar Fair park worked fine at other legacy Cedar Fair parks, without visiting the park it was purchased from, and the add-ons even worked.