That's some pretzel logic there. They can't arbitrarily implement/enforce rules without first disclosing them, and it looks bad in a court of peers if they start doing this haphazardly. I would actually say it's easier for them to get away with this in an at-will environment without these rules.
"Position was eliminated" works pretty well (in the US).
The goal is to keep it out of court or some other tribunal.
Without a contract, discrimination law is the sole course of action to fight an adverse employer situation.
Eliminating a position leaves a company open for problems, particularly if it isn’t true. Much easier to determine that you didn’t charge vacation time correctly or violated a series of minor/moderate rules.
Unless you also happen to be recruiting for that same position, or if your local legislation asks that you prioritise relocating your current employees in other jobs over firing them if their position disappears.
It’s great that you think that, but what I described is what the government does (because of civil service protections) and what every sane HR department with a high legal risk workforce does. Any US employment lawyer would advise that you follow this practice.
What world do you live in where anti-discrimination laws aren't already aren't already utilized by the employers you've referenced? It's really hard to get fired as a US Federal Government or Union employee (based on what I've seen from both in a Midwest state, for United Auto Workers) and the EEOC has had historically strong support for employees for the entities I've cited.
Private entities will do what they've always done and my original comment succinctly describes the default outcome. Nothing has changed.
"Position was eliminated" works pretty well (in the US).