I successfully went from E3 to Green Card without getting a H1B first. It's a more annoying path (there was a ~6 month period in which I couldn't leave the US)

I successfully went from E3 to Green Card without getting a H1B first. It's a more annoying path (there was a ~6 month period in which I couldn't leave the US) and riskier (more ways to end up rejected). That said the H1B route would have taken significantly longer even if I won the lottery.