"Min renkontas ili kaj ludas"

You can't jumble it that much. The "kaj" is there to basically join two different things, "ili renkontas min", and "ili ludas". My Esperanto isn't great but to me that sentence sounds pretty gibberishy and the "kaj" doesn't do what it should be doing. If you were talking to Esperantists you would probably just hear "Ili renkontas min kaj ludas."

Also as far as I'm aware, the ability to jumble it is to make it easier for people who speak languages with different word orders and I don't know of any example where someone would feel more natural with a totally random word order.

Be careful how much you switch the word order around because "kaj" can mean both "both" and "and", so "I like both him and her" would be "Mi ŝatas kaj lin kaj ŝin". You only need to worry about this though if you want the meaning to be as close as possible because those two meanings are basically the same.

“Kaj” like “and” in English links together two things. You need to know which these are and therefore you cannot move “kaj” to the beginning of the sentence.

When you do that you need another “kaj”. For example: Kaj mi kaj vi ludas. But here “kaj … kaj” means “both … and”.