Well, if you're saying that offering registration and a time period to become legal is "rewarding lawbreakers," what's your solution now? Round them up and send them back first?
Round them up, ship them back and never allow them to reenter the country.
"what's your solution now? Round them up and send them back first?"
There is no solution that can be applied now to an already large problem that will not raise loud objections from somebody.
The most efficient way is to start NOW with ENFORCEMENT by making both the employers and the workers pay a penalty for skirting the current law.
You have to make the penalties for hiring of illegals severe. And you have to have random audits and inspections of EMPLOYERS, conducted by teams of enforcement agents. This will cost money, employers will hate it. Illegal workers groups, and Mexico will protest.
Im talking severe penalties, such that a mid-large company would not want to take a risk. Still, you may not get to the illegals working for small outfits.
For an illegal, the penalty is that anyone who is caught in this country illegally is deported immediately, and is forever barred from entering legally later in a guest worker program. This is enforced by means of fingerprints and photo ID, which is entered into a database.
If this sounds draconian, try getting a top secret clearance to work on a military program. All illegals who are caught should be fingerprinted, and photographed before deportation.
Require that people at hospitals WITHOUT INSURANCE who do not speak or understand English present valid proof of citzenship or visa status at time of treatment. If they do not have proof, they are treated, then released to LEO /immigration for deportation, same rules apply. Of course, some will get fraudulent papers on the black market. Hospitals will protest, so federal funds will have to be provided to hospitals to hire additional staff to enforce the laws. Some municipalities will not want to comply.
Hopefully, the result of these actions is that many illegals will voluntarily leave this country if they can, and start the legal process to register.
Now we get to enforcement at the borders, to insure that these people cannot slip back in later. This is where technology that we currently have is deployed efficiently.
All the steps the feds take must be broadcast loud and clear to the Hispanic community, and to the Mexican government.
Everything above is to START.
With the above in place, then you can start to implement a guest worker program. Otherwise, its the cart before the horse.