If we read early works (from the late 80s) in what is now called "deliberative democracy", we can see that it began as a response to certain models of democracy that emphasize preferences and procedures -- up to and including the participatory models of the 70s and early 80s. Although deliberative democracy is not a direct response to Arrow's impossibility theorem in particular, it was intended to sidestep its troubling implications as well as other problems with the older models.

There are two major factions within the deliberative democracy camp. One is indeed interested in consensus building and eventually-consistent rationality. This is the "Rawlsian" faction led by Gutmann and Thompson. The other faction, however, focuses more on actual practices of negotiation through which pre-existing preferences and power structures are transformed. This is the "critical theory" faction led by John Dryzek and the late Iris Marion Young. Personally, I think the latter remains closer to the original aims of deliberative democracy and presents a better contrast to the older models it was intended to transplant. The Rawlsians just took the opportunity to cram their own agenda into democratic theory, as they always do with everything they touch.

Your claims (2) and (3) seem to contradict each other. If it is so abundantly clear that a procedure that satisfies Arrow's conditions is desirable, why do you say that Arrow's theorem is just a mathematical result that doesn't explain anything IRL?

Arrow's theorem is surprising and troubling only if you believe in some sort of sacred relationship between the trinity of democracy, voting, and satisfaction of all pre-existing preference. To the contrary, I find it both trivial and intuitive that it is impossible to satisfy all of the preferences of all human beings, and I would be very surprised and troubled if someone claimed to be able to do so.