People understood that heredity was a thing — that offspring often (but not always) resemble a "blend" of their parents (the "not always" referring to "sports" like the proverbial black sheep that seem to defy the "mixing" principle) — for the reasons you indicate (it is impossible to miss if you deal with agriculture or even human families). But the mechanism wasn't clear. They didn't think about it in terms of "information" (even the early geneticists did not conceive of it in that way) like we do today, nor "God did it" or "cosmic coincidence." Rather, they seem to have understood it as "just one of those things": children resemble parents, plain and simple, just like the Sun rises in the east and sets in the west.

Efforts to try and come up with the "law of nature" that underwrote this phenomena are relatively modern. What is interesting in hindsight is that the approach we take to this today — that heredity is about the passing of some kind of information that is interpreted by cells — is not at all how people thought about this until relatively recently, and that even people like Darwin didn't really think about the problem in this way. Rather, from Aristotle onward the problem tends to have been framed as a question of development, e.g., how do you go from an acorn to a tree, from an egg to a chicken, from a cell to an organism (a conception that only emerged after the microscope was invented). So it's less about the moment of conception and what went into that (which was always viewed somewhat vaguely), and more about how the organism itself develops and grows.

This is what eventually, post-Darwin, got re-framed a "genetic" conception that the real issue is not development but the combination of things (conceived of as particles or "unit characters" and later "information") that go into the creation of the organism. But this requires a very different conception of even what biology is than the one that came before it.

Consequently, this is one of those questions that seems sensible but is actually ahistorical, sort of like asking, "what did people think about the surface of the Moon before Galileo?" — in general, they didn't think about it, and the few who did never took it as a key problem to be solved. What is remarkable to me is that two of the major figures who we now credit for reframing the problem — Mendel and Darwin — viewed it in an antiquated way as well, but their work ended up getting absorbed into a new way of thinking about it (genetics) that ended up radically shaping how later scientists saw it.

The philosopher Thomas Kuhn is notable for his idea of "paradigms," which are not just your worldview, but your metaphysics that goes into it: your idea of what is a worthwhile question to ask, your idea of what the problems or puzzles to be solved actually are. While there is much one can critique about the specifics of Kuhn's approach, it does provide a very compact answer to this kind of question: the problem of "genetics" just didn't exist for the most part prior to the late-19th century, in the sense that it wasn't understood as a distinct thing that one ought to be trying to "solve."