The rendering engine part is kind of the least problematic bit, because there are three primary options (Gecko, WebKit, and Blink) and they're all pretty up-to-date and open source. You could also pick up Goanna, KHTML or Servo if you didn't want to use any of the big three as they're open too, but there'd be work to do on them.
There's no particularly good reason to start your own renderer from scratch.
I'd agree that there's no good reason to start your own renderer from scratch, but I'd argue that there's a great benefit having a competing rendering engine (in this case, Gecko). If Firefox (and, by extension, Gecko) were to die, or fall into obscurity, there would be nothing stopping harmful proposals from being added to various web specifications[1], which is what I was getting at.
There's no particularly good reason to start your own renderer from scratch.