The assumption made by many in the early 20th century, spurred on by the recent successes of unification and formalization, was essentially that we could formally describe the entire universe. Godel’s proof shows that if you attempt to formally describe something there’s either an inconsistency or it’s incomplete. That doesn’t mean you cannot describe the behavior of a dog formally but it does mean the same formula which encodes the behavior will either be inconsistent or incomplete. It might only be inconsistent or incomplete when applied outside of defining the behavior of a dog though. That’s why the little preamble about unification exists in this post but it’s not very well tied into the rest of the post.

>This is what I don’t get. Was this thought impossible before Principia? On what grounds?

Even the full formalization of mathematics wasn't considered certain (*) before Hilbert/Principia and those 20th century attempts. Much less the formalization being applied everything in the physical world or even animal behavior.

* in retrospect rightly so, because it wasn't, at least not without inconsistency/incompleteness.