It's not any more or less out of reach for a medium-sized corporation than it's been since the 70s. The reason no corporation has gone is that there's no economic incentives to. And there's still not; this is a NASA mission, it's exploratory science funded by the public.
How do you expect to get to the Moon before SpaceX? The Space Shuttle ended up costing $2+ billion per launch, and given it's a government program - getting anything on it from an untested company who didn't already have major connections would be near to a nonstarter, and that's even if they could casually foot the billions of dollars it would have cost.
FireFly launched as a private company on a Falcon 9 so their cost was probably a peak of ~$0.07 billion (to maintain units) which may have been able to be privately negotiate downward given the nature of the mission.
The space shuttle isn't relevant to this conversation. Private industry has been putting things in space for decades, long before SpaceX existed. The reason no private company put a thing on the moon isn't because they couldn't, it was because nobody was paying them to, and because there is not otherwise any economic benefit to them for doing so. If your expected income is zero, it doesn't matter how low your costs are.
I agree that Space Shuttle is not relevant (as a peer poster mentioned it simply lacks the delta v to even get to the Moon), but do not agree that this capability has readily existed for decades. You needed (1) private industry (2) affordable (3) moon capable and (4) civilian availability. We were failing on various points at various times, but now a days none of those really pose an issue anymore.
Now that we've clearly nailed all 4 of those points, we're starting to see lots more interesting things in space from tourists visiting the ISS, people sending their ashes to space, to doing private space walks and going further into space than any human has since the 70s [1], and now even things like this with a private company landing a payload on the Moon. Many of these things are done with no return beyond doing them.
More specifically though, this is also literally why SpaceX was created. Elon was researching NASA's plans for getting men to Mars. They literally did not exist. He wanted to get society more interested in space and so his idea was to launch a greenhouse to Mars and live-stream it. The capability for that simply did not exist in America, and in Russia the costs were far too high. So SpaceX was born.
Smallest lunar lander so far is 200kg/440lbs, that's the weight of a carry on full of lead. Besides the Space Shuttle can't go higher or deliver higher than LEO anyway.
That's not an answer as we're looking for choices available from "the 70s" to before SpaceX so let's say around 2000. This issue is literally why SpaceX was started. Elon looked into NASA's plans for sending humans to Mars, saw they didn't exist and wanted to get the public more excited in space. His idea was to use his money to fund the launch of a greenhouse to Mars, which would be live streamed. No company in America was willing/able to do this, and in Russia the costs were far too high. So SpaceX was born.
You're right that Space Shuttle was a nonstarter for technical reasons, but it doesn't change the core issue. Neither does the mass. Unless you can find other companies willing to share a launch with you, you're paying for a whole rocket. And that's if you can manage to contract a rocket in the first place. As an inconsequential nitpick, no lander weighs 200kg. You're conflating landing mass (of which there's been well smaller than 200kg) with total launched mass. Fuel, thrusters, and so on multiply the weight substantially.
