You don't get it, all games on Windows use DXR, RTX doesn't exist as a software implementation. Nvidia did well to make everyone think RT is their invention on Windows.
That being said all games should run on any hardware that supports DXR, AMD or Nvidia. If it works well on Nvidia and it doesn't on AMD, well ... you can use your imagination on why that might be the case.
I don't personally think RT is a Nvidia invention. That's silly. It's been around forever. The main reason it wasn't used was because it wasn't seen as being feasible in real time.
That is to say that I don't disagree with you. There's no reason why it can't be done, as things are at this very moment. Right now, every DX12 game could technically do it and all that they have to do is 'enable' it. And yes, money and markets are definitely a big part. Nvidia obviously wants to have the first and last word, as always. But a big part of my argument is that... they actually can't. For the exact reasons you just mentioned. Unlike some of their other proprietary stuff, the nuts and bolts of their RTX isn't theirs at all. They just focused on building the hardware to utilize what was already there. You could make that case for their older ploys, too. But this is a little beyond those in scope imo.
Right now, it just so happens that Nvidia came around first with the hardware that could somewhat handle it, paired it up with DLSS, and packaged it up in a way that was convenient and lucrative for developers. Those devs can then ride that marketing up and get some fancy new tricks for their graphics pipeline that will at a minimum draw some extra buzz. And when their competitors do it, it ramps up more because it starts to look like you need RTX support somewhere on the box just to sell your game. That's not too hard to see, I think.
But it's not about if it runs. It will of course run on an AMD card, bear minimum. It could run on any card supporting DX12. But will it run well? Seems to be a bit more difficult.
My understanding of the deeper technical side isn't very good, but it seems like it's mostly an optimization game, both with the hardware and the implimetation in the games themselves. In other words: not magic. A lot of tech these days is marketing. But at the end of the day, Nvidia was first on the field for this with special hardware and shiny packaging. But if this stuff is going to have any longevity, that's not going to be able to dominate forever. I wouldn't bother mentioning hardware at all if I didn't think that was a part of it. It really doesn't seem like clever RTX marketing is the only reason RT is less of a thing with AMD cards, even if it's undeniably one of the big ones.
Again, they got there first, so devs look to them first. Simple matter of one is easier to deal with and pays nicely, while the other takes work to get running decently and offers little other incentive, hence it takes longer. Unless you're implying that getting it running acceptably on any hardware is really plug and play, requiring no considerations not already made for RTX. Feel free to correct me there, I don't know enough to say if its really that simple or not.
I don't think RT is enough in its current state for devs to really want to pursue DXR on its own. It's cumbersome. I think Nvidia probably saw that too and started wondering how to get devs to use it more. They saw an opportunity to position themselves a bit higher, took advantage of people's ignorance, and sweetened the pot for the guys who'd wear their new badge. Probably saw it as a good selling point, in lieu of nitty gritty performance gains.
Still, we might not even be talking about RT or DXR if not for RTX, love it or hate it. I personally don't love RTX, but I feel like I do have a decent grasp on why things are how they are right now. I just tend to see it as a necessary evil where one of two things happens. It could fade completely as a gimmick as DXR support (RTX or otherwise) as a whole wanes, and none of this really matters. Or that door opened by the RTX marketing and the push showing that RT actually is now viable eventually spurs enough interest to make it a widespread, staple feature, RTX card or not. If it is good and workable... and the ultimate impact is good, RT will see more use as hardware across the board continues to inch agonizingly upwards. It won't matter what it is called. Nvidia will probably always call it RTX. And I'm sure at some point AMD will jump into the branding game, too. And then one day the branding will be irrelevant because it's just another established method.