The term "Quantum groups" itself, implies that the development of the hopf algebra theory generalizes -in some categorical sense- usual group theory. There are various points that might support this view (although i am not sure if this is what you are really looking for):
- If $H$ is a cocommutative, finite dimensional hopf algebra over an algebraically closed field $k$, of characteristic zero then $H\cong kG$, for some finite group $G$. Similarly, the commutative hopf algebras over $k$ are isomorphic to the duals of group hopf algebras of finite groups.
- The above results can be viewed categorically: there is an equivalence (but not necessarily an isomorphism) of categories, between the category $\mathcal{H}$, of commutative, cocommutative, f.d. hopf algebras over $k$ and the category $\mathcal{Ab}_f$ of finite abelian groups. This implies that in finite dimensions and under the constraints imposed by commutativity and cocommutativity, the hopf algebra theory is "essentially" the theory of finite abelian groups. If we drop commutativity and keep only cocommutativity we have the finite group theory.
I do not know if these are new discoveries, in the sense that they are classical results of the hopf algebra theory; cocommutativity after all is an obvious property in the "tensoring" of group representations. (and of the lie algebra representations as well).
However, -as mentioned in the OP- it is the noncommutative (and the non-cocommutative i would add) aspects of quantum group theory or hopf algebra theory that are really interesting. The notions of quasitriangularity (QT) and coquasitriangularity (CQT), generalize cocommutativity and commutativity respectively. However they still keep close touch to group theory: CQT group hopf algebras are abelian and equipped with a bicharacter $\langle . | . \rangle:G\times G\rightarrow k$. The set of bicharacters on $G$ is in bijection with the set of the homomorphisms of $G$ to its character group $\hat{G}$.
In the f.d. case and for $k=\mathbb{C}$ the complex numbers, the bicharacters of the finite, abelian group $G$ are in bijection with the QT and the CQT structures of the group hopf algebra $\mathbb{C}G$ (that is, its universal $R$-matrices) and in bijection with the braidings of the monoidal category ${}_{\mathbb{C}G}\mathcal{M}$ of the group algebra representations.
In this sense, the non-trivial (co)quasitriangular structures of the group hopf algebra (if the group is finite, abelian these are non-trivial $R$-matrices), correspond to non-trivial bicharacters of the group or to non-trivial braidings of its category of representations.
These notions contribute to the expansion of the definition of quantum groups. Braided groups, are hopf algebras in the braided monoidal categories of representations of (co)quasitriangular group hopf algebras. (i.e. group hopf algebras equipped with non-trivial $R$-matrices or non-trivial bicharacters of the corresponding group).
Edit: Since the OP cites generalizations of group theoretic results to the quantum groups/hopf algebra setting (like the Peter-Weyl theorem), maybe it would be interesting to mention results on the generalizations of Frobenius-Schur indicator for compact groups: In arXiv:math/0004097 [math.RT], the Frobenius-Schur theorem for finite groups, is generalized for semisimple hopf algebras over algebraically closed fields of zero char and to semisimple/cosemisimple hopf algebras if the characteristic is greater than zero. Some more recent results are presented in FSZ groups and Frobenius-Schur indicators for quantum doubles. There, the authors study the problem of
when higher indicators of the reps of the quantum double of a finite group are all integers.
They characterize this as an
interesting group-theoretic question
and proceed in finding groups which have this property and counterexamples as well.
Concluding, i am not claiming that quantum group theory has answered unsolved problems of group theory but it may have contributed some ideas, or at least some descriptions, or even has posed some questions, of interest to a group theorist.