I am trying to land my Ph.D. internship through contacting professors on LinkedIn. Is this a good approach? If not, what would a better approach be?
I would say email is better, for three reasons.
The first is that email is the preferred, standard way of contacting someone in academia, no matter who you are. Now with covid- everything is through email.
The second is that I do not know of many professors keep track of their Linked-In profile. And if they get a message on Linked-In, they will get a notification through their email anyway!
Lastly, and perhaps the most important, a professor could have a bias after seeing your Linked-In page. This is touchy topic but this has happened to many of my friends before, so I'll leave you to decipher this.