This issue has been raised repeatedly, with at least one person stating that she was told by IRCC (yes, second-hand information) that unless a citizenship application was made, the child would not be in the system, and therefore an ETA *would* be issued. I fully understand that the child IS a Canadian citizen according to our understanding of citizenship law. However, until a citizenship certificate is issued, IRCC has not declared that the child is a Canadian citizen, and that is what matters.
I'm not saying that your suggestion of applying for a citizenship certificate and a temporary travel document at the same time is wrong. It's probably the preferred method. I'm just saying that the statement that IRCC won't issue an ETA to someone born abroad who does not now have proof of Canadian citizenship is questionable.
Anyone can claim to be a citizen of any country, but without proof, what value does that claim have? For example, from my reading of British citizenship law, I was always convinced that I inherited British citizenship. Yet, I was informed a few times by British authorities that I had no such claim, because my father was born in Canada, not Britain. Eventually, though, I found a passport supervisor at the High Commission in Ottawa who took the time to go through the various British citizenship laws, step by step, and confirmed that I am a British citizen. What if I was asked years ago on a travel form to give my citizenship? Should I have included British, just because I knew I was British, or should I have waited until the British government agreed with me?