The Beta gate, so far as I can tell, has been in Antarctica functioning for thousands of years. Why didn't the Goa'uld invade Earth through the beta gate at some point in time? Or am I just misunderstanding how two gates on the same planet work?
One good reason is - why? How were the goa'uld to know, of all the worlds with humans on them, that Earth would advance so far technologically?
The Stargate RPG sourcebook (which I believe at one point was canon but is not anymore) insinuates that the human uprising was part of a long period of internal struggle started by Egeria and the tok'ra. Probably dozens of worlds rebelled. Given that Ra (and the System Lords under him) controlled hundreds or thousands of worlds, if one day they find that the Earth gate is not dialing - so what? They know that humans originally came from Earth, but they have plenty of breeding stock for implantation elsewhere. If the planet is just going to give you trouble, you don't need anything there, and you've already got a civil war going, you're going to want to put resources where it matters.
In fact, Earth didn't advance that far - Tollana and presumably Serita, for example, advanced much further. Langara was almost as advanced, and far more resource-rich. There's nothing inherently special about Earth that they'd know which would make them want to go back. (The exception would be Anubis, who knew about the Ancients, but didn't act until much later.)
Finally, who's to say they didn't try? The beta gate was buried at Antarctica. If the goa'uld had tried to send some soldiers back, they would've ended up like O'Neill and Carter - stuck under a mountain of ice
with an unpowered DHD, and it's unlikely they'd be smart enough to repair it. If the soldiers went through and didn't come back, that's all the more reason for the goa'uld to just write off the world as lost.
Everyone is forgetting the Stargate SG-1 "Demons."
There SG-1 encounters a population of Goa'uld slaves who were taken from medieval Europe by Sokar.
Daniel speculates that they were taken through the Antarctic gate.
So riddle me this: Why were the Goa'uld still harvesting humans from Earth AFTER the Alpha Gate was buried and the Earth Rebellion? Granted, Sokar was a black sheep and already overthrown by Ra, it's possible that he discovered Earth on his own hook and acted accordingly, but the question remains: if Goa'uld of any stripe were able to use the Beta Gate, why did they stop coming? For that matter, why did Sokar?
There must be some aspect of the human rebellion on Earth that kept the Goa'uld away for all those millenia. Otherwise, why didn't Ra just send a few Ha'taks after the Alpha gate was buried? As far as I know, this plot hole has never been addressed, and the Beta gate just makes it worse. We can infer that there may have been something protecting Earth during this time though. 10,000 years ago, some Ancients returned to Earth from the Pegasus galaxy. 5,000 years ago, the rebellion on Earth scared off the Goa'uld (see the SG-1 2-parter, Moebius). After returning from the Pegasus galaxy, some of the Ancients took part in the Alliance between themselves, the Nox, the Furlings, and the Asgard. For some reason, they didn't remove the Goa'uld from Earth, but some settled in and mated with the humans on Earth.
However, for the Asgard to influence Norse religions, it stands to reason they may have protected Earth. This isn't canon and I'm mixing in actual history, so things are going to get weird, but looking at the Wikipedia pages for Scandinavian history, the earliest indications of something resembling the later Norse mythology that has Thor, Loki, Odin, etc. is in the Norse Bronze Age. That period spans 3700-2500 years ago. It's possible that's when the Asgard first showed up on Earth. Maybe they started protecting Earth from Goa'uld attacks then, but that leaves Earth not safe for over 1000 years. Potentially other races in the Alliance protected us earlier, without any indication showing up in the historical record.
We know that putting an iris on a gate will essentially kill a person who tries to enter the gate. I'm assuming that burying the gate will have done something quite similar to this, made anyone who tried to use it die.
Secondly, it seems like it's rather difficult to dial the Beta Gate, and that it's only a set of coincidences that caused it to work right in the first place.
Lastly, we know the DHD was buried rather deep in the ice, so most likely anyone who happened on to the beta gate would have died.
Putting all of this together, it seems likely that no one even knew of the Beta Gate's existence, that traveling to Earth through the stargate was essentially impossible. They wouldn't have known if anyone had succeed in dialing Earth, either way, they were dead.
This is admittedly a short and unverifiable answer, but my personal view has always been that the Antarctic gate remained buried for the majority of the intervening time, until random movement of the ice (earthquakes and whatnot) happened to unearth it.
Under this theory, Jack and Sam would have mistakenly arrived through it within a reasonably short time thereafter (say, a few hundred years).
Recall that this gate was originally unreachable (hence the transplantation by Ra of the Giza gate) so we know that at least for some duration of time, it was literally buried in ice.
This doesn't quite gel with those buried Jaffa, though.