I'd like to expand on Dustin's point.  There is simply no
way to think sensibly about equivariant topology, whether
algebraic or geometric, without taking account of multiple
basepoints.  Even taking account of them one runs into 
subtle difficulties invisible without them (see eg [65]
on [my web page](https://www.math.uchicago.edu/~may/PAPERS1982.html)).  I'll give examples from algebraic topology, 
since that is what I know best, but examples from geometric 
topology must abound, as illustrated in other answers.

Take a compact Lie group, or even just a finite group, and
consider a smooth closed $G$-manifold $M$.  What does it mean
for $G$ to be orientable, and what is an orientation? These 
are seriously interesting questions, necessary to make sense
of equivariant Poincar\'e duality, and they are difficult 
except in the boringly simple-minded case (treated in [53] on
[my website](https://www.math.uchicago.edu/~may/PAPERS1982.html)) when the tangent $G_x$-representation $T_x$ is 
isomorphic to the restriction to $G_x$ of an ambient 
$G$-representation $V$ for all $x\in M$.  Usually there is no 
such $V$, and then I can't imagine answers that do not
use functors defined on equivariant fundamental groupoids 
(which themselves are not altogether obvious to define.) Three references which give rather  different answers to these
questions are [93] and [100] on [my web site](https://www.math.uchicago.edu/~may/PAPERSMaster.html), and 
_[Equivariant ordinary homology and cohomology](https://arxiv.org/abs/math/0310237)_, by Costenoble and Waner.
I actually do not know how to compare these answers or to 
calculate with them. 

Again, while one can (twistedly) escape explicit use of fundamental
groupoids when setting up the Serre spectral sequence with local
coefficients nonequivariantly, one cannot do so equivariantly.

Perhaps invoking equivariant theory is overkill, but the fundamental groupoid is such a natural thing, and so elementary, that it seems a little perverse to try to avoid it!