The "E" in ETA stands for "Electronic", and that pretty much refers to the entire process. Not only do you apply electronically (eg, on the ETA website), but the airlines are responsible for electronically checking that you have obtained an ETA or a Visa.
Presuming that your previous flights to Australia have been from the US, you may recall that whilst you were waiting to board the flight they were calling specific people to the podium for a "documentation check" - these are for the people where they have not been able to confirm that they have an ETA, and they will be required to check that they do have the relevant approval to enter Australia (such as a physical visa).
Once you actually arrive in Australia the Immigration staff are able to see that you have a current ETA electronically, and thus do not need to see any further details from you.
If you don't have an ETA (or a Visa, etc) then you will not be allowed to board the plane - it is the airline's responsibility to confirm that you have an ETA before they let you on the flight, and in general they will be fined if they do not so don't expect to be able to talk your way on!
One minor point of semantics, what you have is not an "ETA Visa", it's an "Electronic Travel Authority", or "ETA". The ETA is specifically NOT a "Visa" as such. The distinction isn't normally all that important, and you'll frequently hear people refer to it as an "ETA Visa", but technically it's not correct. The main place where this becomes relevant is if your application for an ETA is denied and you have to apply for a Visa instead. When filling out the Visa application one of the questions is whether you've ever been denied a visa for Australia before, and then answer will be NO, despite being denied an ETA - because the ETA is NOT a visa!