Why do physicists assume that dark matter is weakly interacting?
The short answer is that they don't assume that.
But among all the proposals that remain for what dark matter might be, weakly interacting stuff is the easiest to detect,1 so that is what is getting the money right now.2
And that is not unusual. The history of missing-mass/dark-matter is one of proposals being made and then ruled out one-by-one, in order of ease of accessibility. WIMPs are just the latest candidate to get top-billing. MACHOs were hot when I was in college but were largely disposed of in the nineties and naughties. Before that, decades were spent with ever improving telescopes in wider and wider bands just ruling out many of the ways that ordinary matter could be hiding in plain sight (gas and dust, mostly).
And there are additional possible candidates in the theoretical catalog. I think that sterile neutrinos and/or axions will be next up if WIMPs are convincingly ruled out.
1 There is a caveat here in the form of sterile neutrinos which are not "detected" exactly but can be deduced by finding the three-flavor mixing matrix to be non-unitary. This is a hot topic again because MiniBooNe has recently announced an improved analysis of a larger data set in which the low-energy excess remains and the $\theta_{13}$ efforts have paid off in a big way so we've close to being able to quantify the unitarity (or lack thereof) of the matrix with some precision.
2 WIMPs in a particular mass range also offer the possibility of explaining additional features of the universe which makes them attractive for a second reason.
It's not just the "look under the lamp post" effect. There's also the "WIMP miracle". A new heavy (i.e. about the mass of the top quark, the heaviest SM elementary particle) weakly interacting particle would have an annihilation cross section of about $10^{-26} \text{ cm}^3/\text{s}$. Very general thermodynamic principles predict that thermal production of dark matter in the early universe could only lead to the observed density of dark matter if the dark matter has a similar cross section. This similarity suggests that dark matter might consist of heavy WIMPS.