I’m confused why it won’t clear an existing infection while still working on future infections.

Here is what I know (which may be limited, I’m not a biologist) and also what I’m assuming:

1) The body apparently doesn’t eliminate the virus on its own when it picks up the virus unvaccinated. I’m assuming that this is because it isn’t registered by the immune system as being harmful, for whatever reason.

2) The attenuated virus in the vaccine would not produce an immune response without the adjuvant, because even viruses that are registered as harmful are not reliably registered as harmful when attenuated. This is where the adjuvant packaged with the attenuated virus comes in - it is registered by the body as harmful, and in its confusion the immune system also adds the virus to the registry.

So, naively, if the immune system previously didn’t register the natural infection as harmful, and if it does register the virus in the vaccine as harmful, why doesn’t the registry entry for the vaccine also get applied to the natural infection, the same way as it does for a person who wasn’t previously infected?

Is there some kind of specificity hierarchy, along with a “not harmful” registry alongside the “harmful” registry, such that the natural infection continues to get its previous classification of “not harmful” because the “not harmful” registry entry is more specific than the “harmful” registry entry? That’s the only explanation I can (naively) think of.

And if that’s the case, could we first wipe out the registry by infecting the person with measles, and then give them the HPV vaccine? Just kidding about this part!

I am assuming they meant it won't clear one strain that you already have but may protect against another one you don't