I can get to what you are saying. Videojs is a leader in HTML5 video and considered state of the art by many though I think it falls short on some aspects especially for iOS/Android and Live streaming. Here is a list that compares some common actors of the market.
After much time playing with the different players available I decided to build my own HTML5 video jQuery based player. I learned so much while doing so and if you are planning on re-using it for your projects it will be much faster to tweak your own player rather than trying to build something up on a player someone else built. Now doing so requires you like JavaScript and are happy to deal with the cross browser testing. This article can give you a place to start. Digging in further would require you take on board the W3C spec.
Live streaming in HTML5 video is limited today: HLS for iOS and Android > 4.1 and coming in fast MPEG DASH (it has a JS lib for live and on demand here). If you want to cover a large audience for live video streaming you still need to consider flash.