The bigger issue is that even if one gets their RN license from whatever State without having to produce a SSN, guess what an employer needs especially if you want to work in a hospital setting? Yep, a valid SSN.
The employer needs a SSN as whatever salary they give to you, the employer needs it so that they can use it for their own internal accounting and for income taxes purposes as one of the SSN function, besides documenting that one is legally able to work in the States.
Hospital sponsorships, what are those now? They have been non-existent since 2009 and will probably stay that way for another decade or two, since the US market is so flooded with more US produced nurses and local nursing new grads who many go a year or two before finding a job.