This is false for H-1B - there is no "labor market test". Such a test is required for the PERM process in which an H-1B visa holder would seek a green card through their employer, although the definition of "prove" is up for debate [2].
H1-B applications must have a Labor Condition Application which requires a prevailing wage assessment and that hiring a foreign worker won't adversely affect US workers.
It is true that the LCA (and PERM!) processes are have grave flaws that leave them open to abuse.
It is true that the LCA (and PERM!) processes are have grave flaws that leave them open to abuse.