The vaccine isn't doing anything in your body to fight COVID-19 infection actively. Your immune system is doing the fighting regardless.

That is not what I said.

“ your immune system may not learn how to fight COVID-19 as well, because the vaccine is present and doing something.”

But the vaccine being present isn't "doing something". The thing it did has already happened.

If you develop covid-19 symptomatic disease after vaccination, you will likely still get natural immunity.

The immune system "doing something" is why you feel sick.

The vaccine fighting the virus doesn't make you feel sick because it's just antibodies clumping on virus particles, preventing them from infecting cells.