https://bipartisanpolicy.org/blog/how-do-undocumented-immigrants-pay-federal-taxes-an-explainer/The IRS set up a system called the ITIN (Individual Taxpayer Identification Number). The idea was that it would be used by legal immigrants who did not have a Social Security Number for whatever reason. However, many undocumented immigrants use it to file federal income and payroll taxes (yes, they pay into Social Security and Medicare and likely never see any benefit from it), especially if their employer withholds those (in which case they can get the appropriate refund). Note that they are ineligible for almost all federal benefits and credits. Many file hoping that it will help them in case they ever try to get citizenship, or at least not get deported. (And then of course there are some who have been in the US long enough to get married to citizens, have citizen children, start small businesses hiring citizens and/or legal immigrants--so far from taking jobs, they've created jobs.)
As for the jobs they do, in many cases they are jobs those who can work legally simply will not do. When Georgia cracked down on undocumented workers, their economy crashed because fruit went unpicked and then rotted on the vine. Farmers have tried offering good wages and benefits to people authorized to work--$15/hr, health insurance, retirement pension--and can't find anyone willing to do the work even at those wages. (One story I heard was of a farmer who hired four workers--three authorized, one unauthorized--and the only one who hadn't quit by the end of the first day was the unauthorized worker.) The jobs they're taking, in many cases, are ones citizens and legal immigrants simply will not do, but are absolutely vital to the US economy.
As for controlling immigrant intake, you're right that every country must exercise control over that. However, it is racist to do so by controlling immigration on a racial basis. (And yes, US society generally benefits from illegal immigrants, who pay into the system but get nothing out of it, which helps keep Social Security solvent.) Controlling immigration on a merit basis is absolutely fine. Controlling it on a racial basis (which, in all fairness, Canada did pre-WWII--there was a list of "preferred races," at the top being the British, then the Germans, and various other European nationalities; Jews were allowed only if they were British, or by special waiver--almost never granted, and as a result Canada took in around 5,000 Jewish refugees during WWII and about 6,000 Nazi war criminals after WWII) is a racist policy.
As for the Australian Constitution being "the same throughout"... that means it's created second-class citizens throughout its existence. A provision always having been in there doesn't make it good. Further, as I've noted, some citizenships are extremely difficult to get rid of (US citizenship especially), meaning that the prospective office-seeker can make every effort to get rid of their secondary citizenship and fail through no fault of their own.
You can also help stabilize the economical and humanitarian situation in the countries they are coming from. Unfortunately when it comes to geopolitics US is usually doing the opposite.
Let's keep in mind that a lot of these people are fleeing gang violence in Latin America. That violence is generally a result of powerful drug cartels, which exist with the power they have because of the "War on Drugs".