Correction: Finite temperature is not the only source of adding smearing/lifetime. One can also get smearing/lifetime from interaction terms, by doing perturbation theory. This is most easily seen if expanding the Dyson-Schwinger equation $G = (G_0^{-1} - \Sigma)^{-1}$, and perturbatively finding $\Sigma$ to some order. It will in general have both real and imaginary parts. It turns out $\Sigma(\omega)$ disappears for $\omega \rightarrow 0$ for 0 temperature, however, which means at 0 temperature there is still a well-defined Fermi surface (quasiparticles with infinite lifetime).

Onto the question. In finite temperature calculations, we often use the Matsubara formalism, which is to say we evaluate $e^{-\beta H}$ or observables w.r.t. it by a path integral method, but since the argument of the exponent is real, the time parameter seen in Feynman path integrals is replaced by $\tau$, which people sometimes call "imaginary time" in this context. A Fourier transform yields a description in terms of Matsubara frequencies, as you write.

The existence of the Matsubara frequencies has little/nothing to do with the quasiparticle nature of the particles, but the (anti)-symmetry of exchange of fermions/bosons (quantum statistics) and can be traced back to how the variables (anti)-commute or not, when performing the Fourier transform.

In practice, if one wants to query quasiparticle poles within the Matsubara formalism, one must compute $G(\tau)$ or $G(i\omega_n)$ and analytically continue this object to one defined on the real-frequency axis, $G(\omega)$, and then make the same prescription as OP notes; the poles correspond to quasiparticles. One can instead compute $\Sigma(i\omega_n)$ and analytically continue this, plugging it into the Dyson-Schwinger equation above. This analytic continuation is a trick, wherein you say "if I know the correlation functions on the imaginary axis (a discrete set of Matsubara points for ex.), then simply by treating the temperature as imaginary, I can get out what they would be for a zero-temperature theory, and regain the quasiparticle poles."