Using $448.0 \times 10^6$~$\psi(3686)$ events collected with the BESIII detector, an amplitude analysis is performed for $\psi(3686)\to\gamma\chi_{c1}$, $\chi_{c1}\to\eta\pi^+\pi^-$ decays. The most dominant two-body structure observed is $a_0(980)^{\pm}\pi^{\mp}$; $a_0(980)^{\pm}\to\eta\pi^{\pm}$. The $a_0(980)$ line shape is modeled using a dispersion relation, and a significant non-zero $a_0(980)$ coupling to the $\eta^{\prime}\pi$ channel is measured. We observe $\chi_{c1}\to a_2(1700)\pi$ production for the first time, with a significance larger than 17$\sigma$. The production of mesons with exotic quantum numbers, $J^{PC}=1^{-+}$, is investigated, and upper limits for the branching fractions $\chi_{c1}\to \pi_1(1400)^{\pm}\pi^{\mp}$, $\chi_{c1}\to \pi_1(1600)^{\pm}\pi^{\mp}$, and $\chi_{c1}\to\pi_1(2015)^{\pm}\pi^{\mp}$, with subsequent $\pi_1(X)^{\pm} \to \eta\pi^{\pm}$ decay, are determined