A LinkedIn message that sounds like you know them
A connection note that starts with "I came across your profile" tells the other person you didn't really look. The fix is boring: give your draft real facts first.
Write down two or three things you actually know about the person. A post they wrote, a talk, a mutual contact. If you are using an AI tool, hand it only those facts and ask for a short draft.
Then edit it down to the relationship you really have. If you have never spoken, don't write like an old friend. End with something easy to answer, like one question about the thing they posted.
More on using AI for LinkedIn networking without pretending.