The best thing about portrait photography is the eyes. When you see the eyes and get them in focus the picture captures me. No matter what I do from wedding photography to event photography, I try and capture the eyes.
I started photography as a young kid with small film camera and around 14 with a Pentax 35mm film camera. I started my digital photography with a Canon 60D that has a crop sensor and have captured many great photos from landscapes to people. I Upgraded to a Canon 6D full frame sensor and despite people saying it is all the photographer, not the camera I have to say, yes the photographer has to be able to see what they want to capture and frame the picture. I do know that I have to do a lot less editing after my shoots with the better camera. Although I still can't afford the bigger and better lenses I can see a huge difference in the quality of the pictures. So yes, equipment matters. You can still get good pictures with cheaper equipment, but when you have better equipment and know how to use it you get better pictures in quality.
I recommend using manual in the camera and off camera lighting, Being able to completely control your settings in light will help you produce the best photos out there.
Yes, changing the settings for the natural light can improve it a little, so can Photoshopping it. For the natural light ones I was attempting to get as much sky showing too. It is dark faced or blown out white skies.
Using off camera lighting can capture everything you see with your eyes and take a cloudy sky and give it more than just a faded look that you get with natural light.
There are times when natural light is much better, however you need to pick the times and locations to make it work.
Any "Professional" photographer who charges clients and never uses off camera light is doing their clients a disservice. Take a wedding for example, Natural light for many things is awesome. What do you do when it is in a dark church or the reception is later after dark. Yes the newer mirrorless Sony's can get the picture, but even then, having off camera lighting make it even better.
If your not using off camera light I would start learning now, it is how to really be a professional photographer who is ready to shoot in any light, any time, anywhere. Assuming you brought your stuff with you.