to come and be there: to arrive at a place or reach a person, so as to be present on the scene
This is a verb of arrival. Its heart is not the movement along the way but the moment of reaching the destination and being present. Someone sets out, travels, and παραγίνομαι (paraginomai) marks the point where they turn up and are now on hand. That is why the Gospels and Acts reach for it so often to move a scene forward: a messenger comes and reports, officers come and search, a crowd comes out with swords, travelers reach the next city.
The same idea of turning up in public stretches the word in two natural directions. When the person arriving is a notable figure stepping into view, the word carries the sense of appearing on the scene, the way we speak of a public figure emerging. And when someone turns up specifically to be at another person's side, it can mean coming to their support, standing by them when it counts.
All three shades share one core: to come so as to be present. Choosing among the English words come, arrive, appear, and stand by is a matter of what the arrival is for, not a change in the word itself.