At first thought I would agree wiring issue, but I got to ask, are we sure hazards worked before?
Just about nobody tests their hazards till they have a problem....
Brake lights working eliminates a whole lot of testing - the vast majority... Fuse #1 in both cases of brakes and hazards, everything coming out of and after the multifunction (wiring, bulbs and all). So that leaves a short bit of wiring to the multifunction. R/WH wire that runs through the hazard relay in the fuse box (like 3 inches of wire) and then runs to the multifunction... another 2' of wire.
I doubt unless some PO really spliced in a world of junk into that you would have a problem (especially since that 2' of wire is not continuously hot, it pulses since it is all after the relay - useless for running a toy/accessory).
I look squarely at 2 things, and both are new parts you threw at it, so most likely new and bad out of the box.
Everything I have highlighted green is basically proven good at this point with your brakes working (including one straight through path inside the multifunction that doesn't have any components involved - i.e. switches).
I would start by finding the W/R (white/red stripe) wire coming into the mutifuntion switch and A) checking it didn't de-pin itself B) probing it with hazards on.... if you have intermittent (flashing) power there in normal hazard speed, you have verified the hazard relay and can replace the multifunction. If not, then replace the hazard relay on the backside of the fuse block and test again.
Since turning hazards on kills brakes and inside the multifunction is the only place the two meet I think you will be replacing the multifunction.
Another possibility is that something is shorting to ground in the hazard path - but that should pop fuse #1 immediately.
When you turn either turn signal or hazards on separately, do you get indicator lights on the dash?