Why are vaccines required for our body's immune system to destroy viruses that cause the likes of Covid-19 or Polio, while viruses that cause the common-cold are self-limiting (go away on their own)? What is so different about viruses that cause diseases requiring vaccines?
There is not really an inherent difference between these viruses. Polio is a run of the mill enterovirus - some of its closest relatives are "common colds", while others are less pleasant - see https://www.slideshare.net/TarekMahbubKhan/enteroviruses-and-polio for some examples. Note that some "common cold" viruses have uncommon but nonetheless devastating polio like effects; a leading culprit for acute flaccid myelitis (polio-like paralysis) was the common cold enterovirus D68 ( https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(18)30094-X/fulltext but I haven't checked for updates lately). But for that virus, damaging replication in nerve tissue is at most a rare complication rather than a frequent one.
Coronavirus has long been known for several strains of "common colds"; they were just not as frequently deadly. Unfortunately, we did not see the "cold and flu season" as a demonstration of defenselessness against viral outbreaks, and did not use the opportunity to practice exterminating them.