>A cursory search shows that "dark matter" hasn't been observed at any point in time. It looks like it's a sort-of mathematical placeholder for the "something" that's missing that we can't see (or can't see _yet_.)

Define "see" - that's the scope of the problem. We can see the effects of dark matter. We can measure the effects. What is or isn't seeing is hard to define on this sort of scale.

>Am I correct in my understanding that these alternative theories are trying to prove that there's a way to explain these discrepancies without inventing something unobserved?

More or less.

>Why is that so controversial? To me, again as an outside observer, it feels so counter-intuitive to _invent_ a new type of matter you can't observe than to just say that your calculation is close but not right and to start over. Is it not a crutch?

Because despite many thousands of attempts to provide alternatives, none have passed muster. We couldn't "see" air for quite some time - did that mean it wasn't there? Would a theory positing its existence be a crutch?

No modified theory of gravity has come to close to explaining things like galactic mergers, dwarf galaxies with huge dark matter content, or the galaxies without dark matter content.

You have a situation where we can see gravities and measure their mass. We can figure out the non-dark matter content of galaxies. And galaxies react in a way that really only makes sense if there's additional matter we can't detect - what theory of gravity will explain why two galaxies with similar amount of observable stuff in them have massively different masses? Is the most logical explanation not that there is more stuff in them that we can't see, representing the additional mass? If I have two boxes and they look to be the same size, but when I go to pick them up, one is significantly heavier than the other, the logical explanation is there is more mass inside the other box, despite being unable to see it.

>If I were to take this approach into another field that has an equal amount of controversy where there is a huge gap in my proof, and say that the cause is an "unseen force" that I just invented to explain how my theory works despite the fact that I can't prove in any way that the key part of my theory exists, would I not be laughed out of a room? How did dark matter become accepted while being invented instead of observed?

It's really disingenuous to act like people did a bunch of math, it didn't work as expected, so they just made up some stuff to make it work. The fact of the matter is, you would be laughed out of the room if it was that simple. There has been a massive amount of work done to attempt to prove and disprove dark matter, and there's a reason that it's the most commonly accepted explanation for things. There's also a reason it's one of the theories that people have spent the most time attempting to come up with alternative explanations - and none of them to this day have been able to explain things.

Dark matter might not exist. But it's the best explanation we have at current. People should keep trying to come up with alternative explanations, and continue trying to prove its existence. That's how science works. But you can't expect an incomplete explanation that fails to account for a wide variety of other factors to dethrone one that does account for all of those factors.