We know two versions of Seifert-van-Kampen theorem, one for fundamental groupoids and the other for groups. How do these two relate to each other? I know that the case for groups can be derived from the case for groupoids by treating $\pi_1$ as a groupoid.
But what does this mean in a "practical" sense? I've seen some cases where people use the following diagram
$$\require{AMScd} \begin{CD} \Pi(S^1) @>{\Pi(f)}>> \Pi(S^1)\\ @V{}VV @VV{j}V\\ 1 @>>{}> \Pi(X)\end{CD}$$
to conclude that
$$\require{AMScd} \begin{CD} \pi_1(S^1, 1) @>{\pi_1(f)}>> \pi_1(S^1, 1)\\ @V{}VV @VV{\pi_1(j)}V\\ 1 @>>{}> \pi_1(X, 1)\end{CD}$$
without appealing to the group version of Seifert-van-Kampen. Why are we allowed to do this? And why do we need to go from the groupoid version first instead of just using the group version?
The only reasons I could think of are: we don't need to take care of base points by using groupoids and the requirements for the theorem are easier since we don't need connectedness. But I'm quite unsure, any feedback would be appreciated, thanks!