My plan for immigration:
1) Open all the borders, don't bother policing them (in terms of immigration law, that is. Physical or contraband security and policing may still be an issue). The following plan allows for such simple legal pathways that few if any people should have a motivation in the first place to do any sneaking or lying.
2) DO however, require strict documentation for driver's licenses (and by proxy essentially road usage), school enrollment, etc. where most of our tax dollars are spent.
3) Anybody who has immigrated more recently than the most recent April 15th is allowed to stay until next tax season. You do have to register yourself though upon arrival. They do not get road/school/etc. privileges yet, though. They ARE allowed to work, but their employers are required to withhold their income taxes directly from paychecks (must assume highest tax bracket, then they get a reimbursement for the difference of actual bracket later if the employee does file).
4) Anybody who files their income taxes who moved here less than a year ago is also allowed to stay for at least one more full year. They still do not get any major privileges, although the requirement of withholding income tax can be waived due to their demonstration of good faith in filing this year.
5) Anybody who files their income taxes who HAS lived here continuously since the last tax year becomes a conditional citizen. They can now get licenses, send their (tax-dependent-only) children to school, vote, and everything else, for the next full year.
6) If at any point you fail to file income taxes, you are contacted swiftly at any known address(es) and offered the option to pay up along with a punitive fine. If you do so (or get on an approved payment plan and make your payments if you are too poor), then okay, forgiven. If you refuse or don't respond at known contacts, then you lose any privileges you might have had previously
7) Whenever a person is located anywhere that involves an identity check (traffic stops, attempting to apply for school enrollments or credit cards or whatever, OR authorized police visits for this purpose to known locations of people who used to pay taxes but stopped), you stand to be potentially deported or jailed for tax evasion depending on circumstances and intent if you are not registered under #3 and have not paid taxes either. You may also possibly may be banned (liable to criminal charges if you come back again) depending on intent and circumstances.
8] Native born people ALSO lose their privileges and face other possible penalties temporarily if they fail to file taxes, although minus the deportation option.
This should pretty much solve all the major problems with immigration, while also encouraging a stronger economy and attracting experience and knowledge to the country (reverse brain drain - which may already be true here, but accelerating it). Immigrants would be stupid not to pay taxes when something so simple can give you legit citizenship -- it just becomes massively not worth it to avoid this. SO they pretty much all would. And as long as they do, we should be happy, because they are paying their share for services and opportunities.
The issues with abandoned minors and crap should stop as well, because you'd have no reason to send unaccompanied minors in desperation if you could just legally move your whole family instead and become law abiding, recognized (conditional on continued tax-paying) citizens.
Undocumented workers should be minimized as well, because by offering an easy and immediate legal path to getting a job (the employee withheld income tax thing), it is much less risky for workers to just follow the law than to continue trying to work under the table. Unlike now, they wouldn't be forced into a choice of "starve or work illegally" but rather "save a couple measly bucks and risk being exiled or work legally" which is a much easier choice.
It's also more logical and merit-based. Being pushed out of a vagina in US territory is not a good indication that you will be a benefit to our society. Whereas paying your taxes is an action that proves responsibility and directly earns the privileges of living here. So switching everybody to that standard makes more sense and rewards productive, good people rather than coincidentally-born-here people.