In order to be employed in the United States, you will require proof of the legal right to work.
As part of your hire paperwork, an employer is required to complete an INS form I-9, which details the acceptable documents providing the means for identifying the individual has the legal right of employment in the US.
You will be required to present these documents to your employer to verify eligibility, prior to being hired.
Althought there are penalties levied by the INS for failure to document an employee, far worse liability might be incurred by an employer who had illegally hired an employee and got involved in either a liability lawsuit, or a Workers Compensation lawsuit.
For most employers, the risks simply are not worth taking, particularly regarding any personal injury claim that might occur.
Sponsoring work visa's is done occasionally, I know of quite a number of individuals who have obtained visa's, but without exception, these were inter-company transfers, bringing employees into the US for a variety of reasons.
Until there is an economic impact to business (and consequently taxes) the government is unlikely to make changes to the existing conditions.