That is also not true (except that it is an up-hill battle -- I agree with that). A "Tameny Claim" is an exception to at-will employment, and the rules around discrimination, harassment, retaliation, and discharges in violation of public policy are different. California follows the "McDonnell Douglas" burden shifting process at the summary judgment stage for identifying where the presumption is, and who has to show/prove what.[1][2]
Ultimately, at trial the burden of proof is on the employee, but the standard of proof is low -- more likely than not, which is basically 50.1%. If the employee survives summary judgment, their chances of winning (or settling favorably) are probably pretty good.
[1] See, e.g., https://scholar.google.com/scholar_case?q=Diego+v.+Pilgrim+U...
[2] The burden shifting process is:
"When a plaintiff alleges retaliatory employment termination ... as a claim for wrongful employment termination in violation of public policy, and the defendant seeks summary judgment, California follows the burden shifting analysis of McDonnell Douglas Corp. v. Green (1973) 411 U.S. 792 [36 L.Ed.2d 668, 93 S.Ct. 1817] to determine whether there are triable issues of fact for resolution by a jury. [Citation.] In the first stage, the `plaintiff must show (1) he or she engaged in a "protected activity," (2) the employer subjected the employee to an adverse employment action, and (3) a causal link existed between the protected activity and the employer's action.' [Citation.] If the employee successfully establishes these elements and thereby shows a prima facie case exists, the burden shifts to the employer to provide evidence that there was a legitimate, nonretaliatory reason for the adverse employment action. [Citation.] If the employer produces evidence showing a legitimate reason for the adverse employment action, `the presumption of retaliation "`"drops out of the picture,"'"' [citation], and the burden shifts back to the employee to provide `substantial responsive evidence' that the employer's proffered reasons were untrue or pretextual."