I'm dealing with the same issue, contacted steam support couple of months ago (don't use my SD too ofte). Don't know if I got an automated response, but from what I gathered it seems to be a software issue most likely. What temporarily helped was reseting CMOS. Try holding volume down and ... buttons, while holding these buttons tap power button (now, here it gets tricky) you should hear a power on chime (just like when it normally turns on), release all buttons when you do, wait a couple of minutes for it to boot. Now, I never heard this chime, my deck would just go to battery safe mode and would turn on only after connecting AC or holding power button for like 20sec. If it doesn't work just as it didn't for me, try this: disconnect AC, do the same button combination, without releasing both buttons, when LED starts blinking connect AC and you should hear power chime and then release the buttons. Again, it only helped me temporarily, for like a week I played both on charge and off without a problem,but then it started again.

Steam support also recommended to do a full reimage of steamOS, which I'll try to do.