Question: How do we know that dark matter is dark, in the sense that it doesn't give out any light or absorb any? It is impossible for humans to be watching every single wavelength. For example, what about wavelengths that are too big to detect on Earth?

Answer: 
There is indeed very good reason to believe dark matter is dark - apart from all the evidence from "missing mass" in luminosity counts and gravitational lensing studies.

This comes from theories of large-scale structure formation:
That there has always had to be some sort of matter that doesn't interact electromagnetically at all is crucial to most scenarios of large-scale structure formation. The density fluctuations in the present universe would be too large than what would be predicted if there were only ordinary baryonic matter that interacted only electromagnetically. With dark matter, you can have something that gravitates yet decouples from radiation much before baryonic matter does. This allows the dark matter to form gravitational wells (under collapse) which have a much longer time to expand with the universe. By the time ordinary matter decouples from radiation and joins the rest of the expansion flow, the ordinary matter will quickly fall into these large gravitational wells of the dark matter that have had far more time to grow. This, in a way, amplifies density perturbations in the early universe and allows large-scale structure to form. (to the extent that we see it today in the form of clusters and galaxies)

The required amount of dark matter calculated, in this way, in order to observe the present scale of density fluctuations matches very well with the amount of dark matter required to explain galactic rotation curves, gravitational lensing, etc. So there's excellent agreement that all of these are due to the same thing - some sort of matter which doesn't interact electromagnetically at all, viz. dark matter.
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> **Note:** this answer is similar to [this one](https://physics.stackexchange.com/a/594505/251244) also written by me. [That question](https://physics.stackexchange.com/q/420288/251244) is very closely related and an interesting read.
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> How do we know that dark matter is dark, in the sense that it doesn't give out any light or absorb any?
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To answer your question, you need to understand how dark matter was hypothesised, so here is a summary:

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> Using supercomputers, physicists were simulating the Big Bang and the formation of the Universe, applying Einstein's theories of special and general Relativity and Quantum Mechanics, experimenting with different variables to try to arrive at a system similar to our world as it is currently.
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> As they experimented, they found that in the simulations generated by the supercomputers, the matter formed attracted each other *too weakly*; matter and gas were flung out too far during the Big Bang and could not "clump" together to form stars or planets.
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> They tried adding some "dark matter"; matter which did not interact with the strong nuclear, weak nuclear, and electromagnetic force, i.e. it only interacted with ordinary matter gravitationally. This "placeholder mass" solved the problem, and the digital model successfully evolved to the system of the cosmos we observer today.
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> The intriguing thing was that ~$85$% (!) of the universe had to be made up of this hypothesised "dark matter" so that it formed correctly.
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**Conclusion:** the universe can't have existed without this mass made up of WIMPs (Weakly Interacting Massive Particles). So let's go look for it!

*Dark* matter is called *dark* because it is *hard to detect*, even though it is greatly abundant. It is hard to detect because it does not interact Nuclear-ly or electromagnetically, and photons (light particles) are electromagnetic particles. Physicist don't assume that it is weakly interacting, it was named "dark" because it is so.