I agree. I think there are ways to do that that could get more support than the way we're currently doing things.

Imagine if we began processing immigration applications at a rate ten- or a hundredfold of what we currently do. Imagine if just about anyone could get in, barring things like people with actual serious criminal records, etc. Imagine if when you got in via that system, you got some kind of long-term resident visa, which required you to pay an additional tax for, say, the next 10 years. Also imagine that this long-term resident visa gave you a path to citizenship, on condition of permanently renouncing all other citizenships you might hold. In other words, imagine that becoming a legal immigrant was far less onerous in process, but slightly more onerous in official requirements.

Such a plan could be framed as encouraging immigrants who want to "put down roots", and that kind of immigration is much more plausibly spun as beneficial, because people who move to a place to live permanently do not want it to be sucky. By making the process simpler but applying clear costs (e.g., extra tax), it also gives people an easy to way to demonstrate that they want to do things the right way.

Also, making the process more straightforward makes it much more politically palatable to deport people who violate it (which will still happen). A large part of the "bleeding-heart" leftist perspective towards immigrants stems from a sense that many people who immigrate illegally do so because "they had no other choice". If the bar to legal immigration is lowered so that it becomes a live option, this argument is harder to make.