My experience so far here is that at a major, no-one seems to ask or care about what visa you're on to be there. The amount of nationalities in airline flight decks here is remarkably high, so no-one is surprised when you turn up and sound a bit different. To be fair, I have a green card now, but the guy who is on an E3 at the same airline has had the exact same experience. No-one asks, and we don't really tell.
The other thing I think is interesting wrt to E3s and a flow program, is the time it would take to flow. What are they quoting right now to flow? 5 years or there abouts? An Aussie has a roughly 5% chance of winning the dv lottery in any given year. Over 5 years should yield a decent chance of winning it. Or perhaps that WO regional may start sponsoring Aussies captains on green cards, the same as CommutAir is doing now. Maybe you meet an American and go down the marriage path. 5 years is a long time after all.
Immigration comes with a rather long list of risks with absolutely no guarantees, but if your gamble pays off in the US you do win big. Point in case, the guy that first helped me immigrate to the US now has a corporate job with an eye-watering salary package. He's just a regular dude who worked hard and took his opportunities.