Applying to Ph.D, how to reach out to a professor with no exactly overlapping research topic with me?
My advice on making contact with rockstars is to have an intermediary introduce you to them. They are, by definition, very busy. (Concert dates, groupies, wild parties,...). It is natural for them to think of you as "a random student with a random research idea" if you contact them directly. They probably get a lot of messages, all of which turn out to be of little value. 

If you can, find a professor in your field, such as your advisor, perhaps, and have them write on your behalf with just an introduction and a short recommendation that your research on "topic" is interesting. It is harder for even a rockstar to ignore a colleague than a "a random student with a random research idea". 

And having a paper published gives the recommender a hook into explaining why the introduction is being made. 

It is even better if your local professor/recommender is known to the rockstar. It is still better to tag along at a conference and get introduced in person (after the current pandemic dies down, of course). 

But once you are introduced you have a chance to deepen the conversation over time. 

But, in particular, don't flood the person with too much information initially. That makes it even easier to ignore you since it takes time and effort to evaluate what you send even to determine if there is anything of value in it. Many will be too busy to want to bother. 



Personal note: It isn't exactly the same situation, but I was once able to get a good student accepted into the doctoral program at Georgia Tech by making a personal appeal to the dean, who was a friend of mine. The student wasn't a very good standard-test taker, so is GRE was a bit mediocre, but I saw his potential and expressed it to the dean. I was later told that my not-at-all-inflated recommendation was important in getting him accepted and he went on to finish and to get hired at a similarly prestigious place.