This has been my experience as well. Coaching is part of your job if you want to be a lead or a manager, but it can be very difficult to coach people who aren't receptive to feedback.

When I started as a lead I devoted a lot of time and energy to someone who was underperforming and difficult to work with (and whose self-perception didn't match reality) when I should have devoted myself more to engineers who were doing well and receptive to feedback.