$\begingroup$

First of all, even phrasing the question as "I don't need group theory if every question that can be solved with group theory can also be solved without it" is misguided. It just so happens that the definition of a group is a natural thing. You might be able to circumvent it sometimes, but that doesn't mean that you should. If a concept naturally suggests itself, then why should one fight hard not to introduce it?

Now, why is it natural? Because there are loads of structures out there in the Platonic world that consist of a set and a binary operation: the integers, the rational numbers, the non-zero rational numbers, matrices, vectors, geometric symmetries (closely related to matrices), permutations,... the list is almost endless. So it makes sense to capture this common feature of so many familiar objects in a definition.

As for applications, traditionally groups were only thought of as symmetries of geometric objects. Even in this narrow context, the abstract framework is useful, e.g. to count solutions to puzzles or ways of colouring a shape. Here is a concrete example of a puzzle that could only be solved and understood in its entirety using abstract groups (since it allowed us to identify a group of symmetries of a certain object with an already familiar symmetry group). It is also mainly in this traditional function, that groups are of paramount importance to physicists.

Of course, the great insight of Galois was that the word "symmetry" shouldn't be understood too narrowly, and since then, groups have completely permeated all of mathematics. Groups describe the complexity of a polynomial, they describe the complexity of a topological space, of an algebraic variety, of a number field, etc. Given a number field, say a Galois extension of $\mathbb{Q}$, pretty much all its important invariants are groups: the Galois group, the class group, the ring of integers, the group of units in the ring of integers, etc. Similarly, given a topological space, you have its fundamental group, higher homotopy groups, homology and cohomology... You can tell your friend that if we didn't have groups, then we wouldn't know how to tell a donut from a ball!

I should probably stop here, since to give a full account of the usefulness of groups, one would have to write a compendium of all of mathematics.