Could variants arise spontaneously in multiple locations?

Reminder – I am brain injured idiot with no experience in health care. I make observations and ask stupid questions.

We’ve seen news of multiple variants of Covid-19 spreading essentially everywhere.

It is perfectly normal for viruses, which replicate like crazy, to experience mutations. Mutations that lead to greater transmissibility are likely to survive. Conversely, a virus that leads to greater lethality is unlikely to survive as it kills off its host.

As more people have the disease, more people end up with natural immunity and the disease has a harder time finding a successful new host target.

Natural selection favors mutations that pass on to others more easily, and selects against mutations that are more lethal. Make sense?

(I am not a biologist – assume I am an idiot who knows nothing about this.)

Is is possible that the mutant variants that are seen also arose spontaneously in multiple locations? In fact, is there something about the particularly mutations that would have made these mutations more common than others?

If this is the case, then travel restrictions do not seem to accomplish much. In fact, it explains why in spite of travel restrictions, these variant always end up the general population before we even discover they have “arrived”.

The stupid question is then, is this why we have variants everywhere now?