Among some "pro" tips shared in my neighborhood for desperate job seekers is the myth that promising to never unionize drastically improves the chance of landing on a job. If one were to actually say that in a face-to-face interview then get employed, are they never allowed to unionize? Does it not matter unless the clause made it to the actual contract?
One problem with your example is that under US labor law, employees always have a right to unionize. We could then substitute a different promise, such as a promise to reside within 5 miles of the factory. If you break that promise while employed, you could probably be fired (I assume no specific written labor contract, just the usual verbal exchanges about pay and duties). However, you can also be fired if you don't break that promise – employment in the US is generally at-will, and firings don't have to be justified. Some jobs have written contracts and restrictions on firings, and in that case verbal interview promises are immaterial, what matters is what is in the contract.
Employers in the US do not have the legal power to forcibly prevent a person from moving outside a particular radius. They may have the power to take action that is against the employee's interest. A contract can limit such actions, for example a "tenure" clause may stipulate limited conditions under which a person can be fired. If the employee engages in fraud in inducing the company to hire him, the contract could then be invalidated. There is a thin relationship between fraud and promise-breaking. A promise is when you vow to act in a certain way in the future. If you make a promise, the promise is material to the employer's decision to offer you the contract, and you do not at that time intend to keep that promise, you have committed fraud, which could suffice to invalidate the contract. The employer might have a hard time proving these elements, but that is at least a way in which an job-interview promise (to do in the future) could become legally enforceable.
If one were to actually say that in a face-to-face interview then get employed, are they never allowed to unionize? Does it not matter unless the clause made it to the actual contract?
It probably does not matter even if the clause is written into the contract, because statutory rights such as the right to unionize typically cannot be waived, and any contractual agreement purporting to waive that right, or to agree not to exercise it, are typically "void as contrary to public policy."
Another thing to consider is the meaning of "not being allowed to unionize." In general, contracts and laws enforce prohibitions by specifying negative consequences. The most severe negative consequences in an employment relationship are things like a reduction in pay or being fired. So you have to compare the employer's capacity to reduce the pay of, or fire, someone who hasn't made such a promise to see whether the promise is effectively enforceable.
This in fact explains why employers can't bind their employees to a promise not to unionize, and why the invalidity of such agreements is "contrary to public policy": the point of a statutory right to unionize is to prevent employers from using their position of power to exploit employees. If employers could use that position of power to extract binding promises not to unionize, that protection would be meaningless.