Enloe11 wrote: so by letting my left arm bend i will drive my chest more, swing longer and faster, and end up on top of the pole?
Yes! Focus on your chest --- not on your bottom arm. To swing longer and faster, you should also do lots of highbar work.
I agree with Charlie. And with Vquest.
You may have to use lighter poles while you transition from a bottom arm press to no bottom arm, but it will pay off in the long run.
The key to a good swing is good posture immediately after takeoff. When I freeze your vid to post-takeoff, I see your body slanted backwards (not forwards). So just as you should have good forwards lean (good posture) when you run, you should have a similar forwards lean post-takeoff. Let the chest drive through, rather than resisting it with the bottom arm. Don't worry, the top arm will bend the pole, due to the force generated from your run and jump.
With good body posture, you will get the kind of PR improvements that Charlie mentioned.
Re head posture, don't do anything with it --- just let it stay aligned with your spine. Some vaulters are under the misconception that if you lead with your head, your body will follow. This isn't true. Arching your neck actually causes your entire spine to arch --- just when you want the opposite to happen. During your swing, you want your spine to become hollow (the opposite of arched).
Sam Kendricks throws his head back, but that doesn't mean you should copy him, just because he's cleared 19-3 this way. I consider it one of his worst technical flaws (and at 19-3 he doesn't have many).
Good luck to you, and here's hoping for many PRs this season. Let us know how it goes.