Some diseases could be eradicated if they were eliminated from every person. But many diseases have reservoirs in the environment (tetanus, Legionella, histoplasmosis) or in animals (West Nile, monkeypox, Ebola, MERS). Other diseases arise more or less spontaneously due to e.g. mutation (feline infectious peritonitis -- I can't offhand think of a human example) or recombination with related animal viruses (SARS, SARS-CoV-2). Since these diseases don't depend on remaining in the affected population, they wouldn't be eliminated by clearing them from that population.

>if we were lucky enough to get to a point where nobody would be infected by smallpox, would that mean the end of smallpox

That's exactly what happened, but there was nothing "lucky" about it - it was the result of a decades-long enormous vaccination and eradication effort by every country on Earth,