In practice the mechanism is that every employer figures it in to their offers, such that every offer you ever receive is ~6.2% lower than it would be if SS didn't exist.
Tax cuts go straight into the employer's pocket, as the employer will happily tell you to your face:
https://theintercept.com/2017/01/05/corporations-prepare-to-...
Sure, it's theoretically possible that the employer could pass all tax cuts to its suppliers. McDonald's could call up and offer to pay more for frozen hamburgers and fries. Walmart could tell its Chinese shoe makers that it plans to pay double for all future pairs, just because. But, why would that happen as opposed to rewarding the shareholders?
Every offer is up to 6.2% lower. I have a business with employees but their pay is sufficiently low (compared to the revenue of the business; their pay is above market for the jobs they do) that SS does not enter into our calculations. If the employer portion of SS was eliminated tomorrow we would not give anyone a 6.2% raise, nor would future employees get offers 6.2% greater.
I'm sure somewhere there is an employee whose salary is a big piece of a business's income and they would have gotten a little more money had that tax not been there. But I'm pretty sure that's the exception rather than the rule, if for no other reason than SS taxes are capped so you reach a point where more salary does not equal more SS tax.
Indeed, social security isn't taxed on anything above 118K.
[EDIT] With health care costs being fixed at crazy high amounts (in the US at least.)
Not quite. This is actually split between the employee and the employer. With each contributing 6.2%.
https://www.ssa.gov/pubs/EN-05-10022.pdf
Also why are you admonishing the OP? He has every right to pessimistic about this current state of social security.