How would Americans go about paying them a non-tipped waged per/hour and allowing the increased wages to be included in the menu prices?

Restaurants in the US that eliminate tipping (few and far between as they may be) implement a service charge (about 18%) on every tab and use this money to help increase their servers' wages.

Can people who do the same job as me in the USA do the same thing?

It looks like that there isn't much real wage difference between US and Australian servers, so the ability to travel comes down to personal lifestyle choices (like being single or not having children) and having the time to travel. You mentioned getting four weeks of paid vacation, and this is definitely not something servers receive in the US. If we don't work, we don't get paid, though it can be easy to find someone to cover your shifts if you do want to travel.

Why is it then that they always whinge about being poor and never seem to be able to pay their mortgages?

If you're talking about this subreddit, it may have to do with the fact that people tend to come here to vent, so there's a certain amount of confirmation bias. But, statistically, tipped employees in the US experience poverty at 3 times the rate of the workforce as a whole.

For perspective, the poverty threshold for a 2-person household in the US is $15,930 USD. If you're a single parent, the struggle is real.

It's also more difficult for tipped employees to qualify for bank loans (including mortgages). Due to their variable income and under reporting their income, banks see them as riskier investments. As a result, a lot of people rent (like I do), and rent is always at a premium.

Conversely, for those working part-time, who started waiting tables as a second job to make ends meet, they're already over-extended on their bills. They rely on the extra income to maintain their family's lifestyle, and not meeting those goals is quite stressful.