This is incorrect, it's "for limited periods of time under certain circumstances" [0] (and citizenship is not required).
> they can always keep their current insurance through COBRA. [...] needs to pay the full premium
This is true but vastly underplays the cost. It isn't unusual for "the full premium" to be over a thousand dollars per month [1] (!). They even get to charge you an "administrative fee". Oh, and there's also tax implications.
> For people who don't want to stay on their previous employer's plan, the loss of the job is a "qualifying event" which allows them to immediately do things like join their spouse's plan or sign up for an individual "Obamacare" plan through healthcare.gov.
More likely for people who can't stay on their previous plan because it would bankrupt them. Or if their spouse is not working. What happens if the employee has children/dependents on the plan? Etc, etc.
None of this is easy. Switching insurance changes which doctors are "in network", so at minimum it's significant hassle. At worst, their or a family member's current treatment is only partially or not at all covered.
So, in practice, parent's description is accurate enough.
[0] https://www.dol.gov/general/topic/health-plans/cobra
[1] https://www.investopedia.com/cobra-insurance-cost-7486539
This is a common misunderstanding.
If an American loses their job for any reason, they can always keep their current insurance through COBRA. In fact, they don't even need to take any immediate action. If they need healthcare in the month of, or month after, their loss of job, they can go get it and then later on pay their premium to have the insurance retroactively go back in time to cover them. The only difference is that the covered person needs to pay the full premium, while the standard employment agreement has the employer paying a significant portion of the premium.
For people who don't want to stay on their previous employer's plan, the loss of the job is a "qualifying event" which allows them to immediately do things like join their spouse's plan or sign up for an individual "Obamacare" plan through healthcare.gov. There are subsidies based on income to make the premiums manageable for low-income people if they can't afford the market rate.