Pages 8 and 9 of them mentioned paper discuss naturalness vs renormalizability, so it is important to clarify the difference.
First of all, it is nice to have a renormalizable theory because you can calculate things. However, it is not that "since we can do perturbation theory without paranoia we expect the coupling to be always small enough" as mentioned in the post, but they other way around. Namely, whenever the coupling is small, we can do perturbation theory and we should aways check this and be able to justify why we think the coupling might be small before we start expanding.
That being said, let's assume we have a renormalizable theory, which will of course only involve relevant opearators at low energies. Then (depending on what we are using the theory for) we might also want to check if it is natural or not, in the sense described by Polchinski.
This is better explained by his example: If we take the theory of a free scalar, then renormalizability allows us to add a term $m\phi^2$ to the Lagrangian. We can choose m to have whatever value we like, but no matter what we choose, there are loop corrections to the mass that will contribute to the value we chose. For a free scalar the corrections scale like $\Lambda^2$ which means that at the end of the day the mass of the scalar will be around that scale, unless miraculous cancellations happen in the corrections among terms of different orders. Such cancellations would be un-natural, hence justigying the terminology. Note also that naturalness is intimately related to hierarchies: it is un-natural to have to different scales $m$ and $\Lambda$ that are very different from each other.
So at this point we have two options:
i) we either include a mass term because we might be doing for example an application in condensed matter physics where nothing forces us to demand naturalness or
ii) if we are trying to study a fundamental theory of nature and we want it to be natural then we need to come up with an extra condition that forbids such terms. In QED, we impose the requirement of chiral symmetry, which then forbids these mass terms.