Did historical polytheists really say things like "By the gods!" as a generic curse or is this a modern invention?

As to whether the ordinary person would have thought of the gods in such vague terms is really a question for another day, since it's quite complicated. Let's just say that what we're taught in high school and whatnot about there being a set of very well-demarcated gods and goddesses with full anthropomorphism and completely separate spheres with well-defined roles is not really true. The reality depends on what time we're talking about or whether we're talking about the Greeks or the Romans.

In Latin the exclamation di immortales was probably the most common. We find exclamations to specific gods, as well--in Latin mehercules, or my Hercules, is probably the next most common. Swearing by Jove is relatively rare, usually reserved for the swearing of oaths, which Jove protected. However, this isn't true in Greek. In Greek, and often in Latin too, we find that people make exclamations to "the god." There are lots and lots of reasons for this, and with a really good grounding in the ins-and-outs of ancient religion makes it clearly why this is natural. Partly it's in case you're not sure which god you should be addressing, or to avert bad luck or the evil eye--lots and lots of reasons. Like I said, neither the Greeks nor the Romans thought of their gods as being as cut-and-dried as we do.

Anyway, I guess I haven't answered this particularly well, mostly since I don't have a catalog of all the exclamations that appear in the textual body. But exclaiming things in the name of specific gods is very rare. Oaths are a different matter, though. Oaths of course would be sworn in Zeus' or Jove's name, along with any other gods you though should be mentioned because they were relevant. Prayers also of course would have been sworn to specific dieties.