If you have the content, it will rank. Bing and g both have large listings of Australian TLD media, etc.

Is it a magic bullet? No.

If you only talking about in Australia, you're fine. If you want global ranking, as said, if you have relevant content you should rank as well as any other relevant content, but there is no guarantees with any of this.
Make sure your country specific terms and relevance relate to the target market.
Example (very obvious) , I would not be interested in mower repairs in Australia unless I was local.

One last consideration is country TLDs generally suggest to the search engines a geo location in the results. To go global there are other domains that are more easily recognized, such as ".com".

Thanks for the replies! Seems like content is king, and especially local content in this context.

More specifically, I am interested in .com.au versus the new .au domains. The Australia Domain Registration has been pushing this as a top-level, higher ranking domain than .com.au. They are both for the Australian geo location, yet they're claiming .au will rank better than .com.au? Would you say their claim is mostly marketing?

New? .au is not some new offering, it is a ccTLD and has been available since the mid 1980s. If you have a business or service doing business in Australia, you would rank better in Australia. It will not likely bring you much new traffic if your business is in another country. It would not make very much sense to buy an .au domain if you are not seeking specifically Australian traffic.

Thanks for the response not2easy! While the .au ccTLD has been available since mid 1980s, it had to be used appended to .com, .gov, .edu etc. Direct .au was made available to the public from 24 March 2022 and so businesses can now register domains such as businessname.au instead of businessname.com.au.

I was wondering if there was any benefit to preferring the domain that omits .com (eg. businessname.au). Apologies for any miscommunication.