Conservatives (the politicians) pretend to want to deport illegal immigrants to get votes. They don't actually want to do it because a lot of businesses depend on them for cheap labor. You don't even need to deport anyone, just start actually enforcing laws that require employers to check employee eligibility. Fewer jobs for illegal immigrants would mean less incentive for them to come. That nobody has tried this shows IMO that nobody (in power) actually wants it. They just want a few examples to grandstand about for votes.

pay a penalty, and stand in the back of the line for American citizenship while they are provided recognized status so they can live in the country legally in the meanwhile.

This is basically amnesty/open-borders. Not arguing for or against, but it is. There is no "line" for citizenship, after five years of permanent residency anyone qualifies. The "line" is for the permanent residency (green card), and you have to qualify to even get in the line, that's why people come illegally. So what's the new qualification? "already in the country"? Then what about the people here 20 years from now? Is it a rolling admission? If not then you have the same problem as before. If yes, then the policy is 'open borders' possibly with a qualification of "can avoid getting caught for X years".

If you don't find open borders acceptable the only reasonable compromise I can think of is a capped category that anyone can apply for, where the cap is adjusted as needed. Somewhat ike the current Diversity Visa (Green Card lottery) which is capped at around 50K per year, though that cap rarely changes.