Your suggestion about carbon is not falsifiable observationally. With real data you can only place an observational upper limit, you cannot measure the abundance is exactly zero. Without a quantitative calculated prediction of the carbon abundance it cannot be falsified. Similarly you can only test direct collapse black holes if you have some way of finding them, their observational properties depend on the formation scenario. You also need the expected number density and redshifts of such objects to reject anything.
Dark matter is also not falsible observationally, every time a supposed DM effect fails to be observed it's just assumed it's darker than expected.
One could re-postulate the theory as the innumerable tiny hands of god pushing on mass in the divinely chosen direction and nothing really changes but the name it theory.
The hands are there, they're just smaller than the resolving power.
At some point it's time to admit fault, but so far that's not happening despite the ever accumulating pile of evidence against DM. For a supposedly mature main stream theory the proponents are surprisingly fragile.