The good reason for adding dark matter, that you say is absent, is one that is more a question of philosophy of science, in that adding more mass accounts for the observed behaviours without changing the known laws of physics.

The known laws of physics have been formed on math that checks out and is consistent with all our other observations, and has made many predictions that have checked out and even formed the basis for technology that we use every day.

The way science works is that we form mathematical models of physical behaviour, we test model against real world data, and if the model is consistent with reality, and predicts further behaviours that we then can test for, the theory behind it holds water and we have something to work from.

If you like, you can think of it as building trust in a model, having courage in a theory isn't a mistake, it's how science has been built. Of course finding the errors and new laws is important, but you have to conclusively rule out the established theory first.

This is how we got to Newtonian mechanics instead of firmaments, elements, worlds of forms and mythologies, and how we got to medicine instead of humours, phlegms, biles and alchemy.

Adding mass that can't be seen preserves the body of theory of the standard model and doesn't raise any questions of why GR/QM work correctly for things like GPS etc.

In other words, dark matter is an answer that doesn't require going backwards.

Saying that gravity behaves differently to what we previously thought means that the standard model is only coincidentally right or only right in particular places, and from there where does the scepticism end? Where do you even start unravelling the tapestry?

Think about it in a diagnostic analogy. If your patient is critically ill, and you don't know why, you will prioritise testing for conditions that fit the symptoms and can actually be treated/cured. Because if it isn't treatable, the truth of what caused it isn't that important.

Occam's Razor as an argument against dark matter, but saying that gravity behaves differently — when our theories do not otherwise predict that it should behave differently — is actually less simple than saying there is more mass than can be detected via EM interaction.

The other point I would make is that dark matter can explain most if not all of the otherwise problematic observations, which makes it preferable over modifying the laws of physics, because as I understand it, doing this doesn't account for all of the problematic observations.

Again with an analogy to medicine, it is less likely to be three unrelated, coincidental conditions in one patient than a single condition if both diagnoses explain the same symptoms.

To frame everything I've said in a medical analogy, we have essentially treated for the condition we think it is, and we're trying to figure out why the condition has presented differently to typical cases, rather than ruling out the diagnosis and saying it is something else entirely — because the treatment is working. That is, the empirical evidence we have suggests that our diagnosis is correct, but we don't know everything there is to know about the condition.

Our GPS works, gravitational lensing has been observed, gravitational waves have been detected, we power our homes with nuclear reactors, we calibrate our most accurate clocks based on quantum mechanics, and so on. Particles we then predicted would exist have since been detected.

The patient had a fever. We treated the patient with antibiotics, and they got better, so we have reason to believe it's a bacterial infection — we just can't see the bacteria in the blood work. So the next logical step is to think of what presents and responds like a bacterial infection but isn't bacterial, or otherwise speculate that we have discovered something that does this, rather than question whether we understand the human body or whether thermometers work.

If in trying to confirm this discovery, we find that actually we don't really understand the human body or that our instruments are broken, that's when we should start looking to re-assess the laws of physics.

The standard model has no useful purpose if we don't place some trust in it to find new things. If we threw out our scientific models every time we encountered something we weren't expecting, we wouldn't make any progress at all.

The only reason dark matter raises so many eyebrows is (a) the popular press just loves to pick at it because it's an easy target with great headlining when your scientists are saying the majority of the universe is "missing"; and (b) because the breakthroughs of today are framed as being incremental compared to the big eureka moments of the 19th and 20th centuries which saw us leap from Newtonian mechanics to GR and QM. But between Newton and Einstein et al, there were centuries of incremental refinements/improvements on Newtonian mechanics and early modern astronomy, so why is there such impatience because we haven't found dark matter in barely a hundred years?

By definition, dark matter is going to be hard to detect because the only means of detecting it is merely enough to suppose that it exists, i.e. it interacts gravitationally but not electromagnetically, so we can only detect its gravitational influence on celestial bodies. Of course, it's the odd behaviour of celestial bodies that led us to suspect that dark matter was a thing in the first place, so this isn't very helpful.

I should think that, in order to prove dark matter exists, we shall have to imagine an edge case of what an extreme concentration of dark matter would do to nearby celestial matter and how that might be distinguished from conventional phenomena. Easier said than done, the universe is full of bizarre phenomena, much of which can be explained by GR and QM, and any remainders probably defy any remotely intuitive reasoning.

Alternatively, we shall have to imagine what phenomena might occur in situations where dark matter is absent, and where something can be modelled mathematically as being conclusively due to a lack of dark matter i.e. if gravity were to be different to Newtonian/GR, the phenomenon would never be seen, we can then say with confidence that dark matter is real.

To confirm it beyond any useful doubt, I suppose we would need to create conditions under which dark matter would form and observe the phenomena that occur, or build some kind of instrument that can detect gravitationally as accurately as we can detect electromagnetically. Again, easier said than done, EM force has a fundamental particle that we know very well, while gravity ... well the jury's still out on that. The graviton even if it were a thing would not be a particle in the same way, you can't have a quanta of gravity, when gravity is more of an emergent property of the geometry of spacetime? You can do the math as though it has a force carrier, but this isn't something that you expect to be able to manipulate as a particle in application.

Anyway, I'm away on a pretty hefty tangent now. The point is that it's more constructive to suppose that there is dark matter, since alternative theories (a) also include dark matter, to a lesser extent and (b) do not account for the observed behaviours without in some ways failing to predict behaviours we know and understand with known physics.