The title is without a phone… number.

Yes, it’s another good reason to set up a backup phone number with someone you trust.

And when we left Australia I really had so much stress that I didn’t think about my Google account. We sold almost everything, including car, appliances, furniture, etc., then packed the rest into storage, had to do some last minute repairs on our house in order to rent it out, go through inspections, cancel all sorts of insurances, phone, internet, this-and-that, then 6 days before leaving somebody crashed our second (already sold, but not yet transferred) car, we had to deal with insurance (who first wanted to pay too little for the written off car), cancellation of registration, towing bill, and so on – the fun when one ends his/her stable life to hit the road!

No, I didn’t have any back-ups! As stated, I’m still using a non-smart phone from 2007, I only just entered the world of Androids with a Nexus 7, which I use for website testing and mostly as a GPS when travelling.
Before I last returned my laptop (when it was finally replaced under warranty) it didn’t hold USB connections anymore [all 3 ports], I guess due to a failure of the motherboard. You plugged in your external harddrive, it recognised it, and then lost it within maximum 3 minutes. Hence I couldn’t run a full back-up, only copied the most important directories over to my external drive (and forgot some of them too).
Anyhow: without back-up codes in paper form you’ll run into the same trouble if your smart phone develops problems! You can only use Authenticator on a new device if you are able to transfer it from the previous device!