Note: this answer is similar to this one also written by me. That question is very closely related and an interesting read.
Why do physicists assume that dark matter is weakly interacting?
They don't assume that.
To answer your question, you need to understand how dark matter was hypothesised, so here is a summary:
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.
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.
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.
The intriguing thing was that ~$85$% (!) of the universe had to be made up of this hypothesised "dark matter" so that it formed correctly.
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. Physicist don't assume that it is weakly interacting, it was named "dark" because it is so.