Out of curiosity, why would you convert to an H1B?
Other reason 1: You can transfer to another job easier while remaining in the US (if they do an H1-transfer). Doing so with the E3 is a huge grey-area with various blogs saying different things.
Other reason 2: If I get fired I have a few days to pack my life and leave the country. With a H1B I recall the period being longer to find another job.
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.