The big question is, are you "speedy enough?" As in top end speed. Can you run fast enough that speed is never a concern during racing and training? Only you can answer this. Silly examples, but if your goal is a 3 hour marathon and you can run a 12s 100m, speed isn't an issue. If your goal is a 2:00 800m and you can only run a 14s 100m, speed is a HUGE issue.
If you think you need more speed, gaining strength can lead to gaining power, which directly supports speed. You can, of course, make some speed gains without increased strength gains. It depends a lot on who you are and what you race. An speedy 400/800 guy can typically do some good, heavy squats. A marathoner, no so much.
On the subject of injury. Weight training, when done properly, is significantly safer than distance running. I firmly believe everyone needs to know how to squat, lunge, and hinge properly—heavy weight is not a requirement. These are fundamental movement patterns that will help (help, not totally prevent!) keep you injury free for life. But if you're hips can't hinge right like they're meant to, something will go wrong eventually. It may not be during your running career.