As a father and primary breadwinner, a big risk with working under the table is being injured on the job (never mind the lack of SS contributions you mention, or the lack of unemployment insurance). Here in NJ the underground economy is thriving with the illegals working under the table; Americans simply aren’t offered jobs in some sectors. The owners that use the braceros defend it by saying they can’t compete without them; they’d be underbid on any work by competitors that use them.
My brother-in-law* employs about 20% of Montevideo, illegally of course, for his contractor business. Recently one of them supposedly was injured. The illegal knew immediately to lawyer up and is suing him and now my BIL has some ‘splainin’ to do with a couple state agencies.
*I haven’t talked to him or my sister (a public school teacher) in years. I was told this happened by a relative who still lives in the same city that they do.
And you can probably guess which political party they both worship.