I know that in the pure algebraic context, connected groupoids are equivalent to groups. Moreover, connected groupoids can be seen as action groupoids. Consequently, any abstract groupoid is essentially a disjoint union of action groupoids.
Therefore, is it not more natural to study group actions than study groupoids? Since group actions seems to have a richer structure than groupoids.
So, why study groupoids?
Does this also hold in a continuous context?