1200W should be plenty for 6x 1070s. My rigs are setup like that and with PL of 65-70% pull 775W at the wall and the psu doesn't even turn it's fan on most of the time. If his psu kicks the bucket there's something wrong with it!  80% utilisation should not tax a decent gold or platinum psu.

I'd recommend unplugging all the cables and every riser, check them carefully and then add the cards back one at a time, mining on each for 1/2hr or so at stock before adding another. That should at least let you find out at what point things get unstable. If it gets unstable with one card then look to the software, reinstall from scratch or take a known working image from another machine if you have one.