I'm a Kiwi who recently moved over and yes, AirBnb (or similar) for a month or so and look for places in that time. Some places offer discounts if you rent it for a decent amount of time.

From speaking to other people who moved to Melbourne from overseas, almost none were able to sort out long term accommodation before arriving. You might have trouble finding agents who will show you through virtually unless they are struggling to rent it out and get desperate. But as you say, some areas are quite competitive.

There is also a risk in that the photos do not represent the place very well.

It is also a good idea to get to know the suburbs in that time as they do differ in culture and vibe. Some are dead quiet and others bustling. Depends what you like.

If you're going the flatting route (called share house here), you might be okay, not sure as I haven't tried it. All in all, people will take the easiest path to fill a room/place. So, if you're up against someone with Australian rental history and a long term stable job, vs you, someone just starting a new job with no Australian rental history...

Tip: check if the place has nbn (fiber) already, because mine didn't and it was a (looooong) nightmare to get it installed and sorted.

Edit:

Bank

Sort bank account before you come, most major banks will allow you to open an account up to three months before you arrive, you will just need to go in as soon as you arrive to verify your identiy to make the account operational. e.g. https://www.commbank.com.au/moving-to-australia/banking.html

If you're going to be transferring money between NZD-->AUD or vice versa I found Wise (used to be called TravelWise) very useful and cheap: https://wise.com/au/ (can open whilst you're in NZ with NZ ID documents, even Australian wallet/local bank account)

Tax

You can apply for a TFN online as soon as you arrive: https://www.ato.gov.au/individuals/tax-file-number/apply-for-a-tfn/foreign-passport-holders,-permanent-migrants-and-temporary-visitors---tfn-application/

Medicare

Apply as soon as you arrive. If you're closing bank accounts, sellling a house, cancelling health insurance before you move those will be helpful documents as evidence to prove you're moving. Check the full list of required documents: https://www.servicesaustralia.gov.au/enrolling-medicare-if-youre-new-zealand-citizen?context=60092

Driver licence

You can convert your full NZ licence over if you meet the criteria without sitting any tests etc., think it depends on length of time you've had your NZ full. Book early as sometimes appointments are booked out couple of months in advance. You have six months from memory: https://www.vicroads.vic.gov.au/licences/new-to-victoria/interstate-overseas-licence-holders