Read the first paragraph of this Wikipedia article carefully:I've learned not to get into pissing contests with you, but I do wonder if you know the real purpose of a lightning rod. Hint: it isn't to attract lightning strikes.
https://en.wikipedia.org/wiki/Lightning_rod
Pay special attention to the sentence that includes the word "preferentially". That means that if it is going to happen, you prefer that it strike your lightning rod system. You're essentially making your lightning rod more "attractive" than what it is protecting.
Whether an antenna is a lightning rod or not, if it is hit, there will be extreme voltage on the coax so the best you can hope for is to reduce the chances of the antenna being hit. To that end, you don't want to make a lightning rod out of the antenna.