As you lift weights, run more, and run faster, running form improves naturally for many people. For those whose form doesn't improve, they get injured so often they quit the sport. Thus, those who say running form improves naturally are looking at a self selected sample of themselves and others who became good without any conscious effort to improve form. They (the people on this forum) are actually the minority of people who run. Many beginning, high school, and recreational runners could benefit greatly from improving their running form. If you doubt this, watch runners finishing from 20 minutes on back at your local 5k. You will see lots of form flaws that increase the risk of injury and limit the runners' potential to get faster.
I think your coach's explanation is confusing. You don't actually push from the ankle. Your ankle never touches the ground. To me, running from the ankle implies that the foot stays dorsiflexed throughout the entire foot strike. It does not. The foot is dorsiflexed at landing and plantar flexed at push off.
You actually push off from the forefoot, but it's too simplistic to just say "push off with the forefoot." "Push off" is a coordinated effort of driving the arms, extending the leg behind you, pushing off with the forefoot, keeping your pelvis tucked in so you get slightly more knee lift naturally with the forward leg, and slightly more forward lean. Oh, and keep your torso aligned and not bent forward. All of this together leads to a better push off.
You might be good at parts of the above and weak at others. We don't know. We haven't seen you run.
There's a lot more to good form. The above comments only address push off.
Lifting will help. Running drills like skips will help. Looking at slow motion videos of your own running form will help. Watching slow motion videos of top Kenyan runners will illustrate all the form components I outlined above.
Again, I haven't seen you run, but it's easier to coach some runners by showing them how to adjust their arm swing. For example, if you concentrate on extending your opposite elbow farther back, your trailing leg will also extend slightly farther back, giving you more of the push you are looking for. It's worth a try, but changing armswing is difficult for many runners. There are drills specifically for armswing, but that wasn't your question and this post is already too long. Good luck.