2011 MBP boot problems

2011 MBP original owner, 15" 2.4 GHz core i7, mods include replaced battery and 2 Samsung EVO 1TB SSDs.

The logic board was replaced by Apple for free in 2014 due to the well-known video issues, and ran Mavericks flawlessly until the middle of this year when I upgraded to El Cap then Sierra, which ran fine for ~2 months then boot problems.

Now it won't boot, in any mode - Internet Recovery, Recovery, Safe, Verbose reboot, external boot drive all fail partway through the progress bar evolution. This started 2 months ago as an infrequent thing, became more frequent, and now it won't boot at all. Resetting SMC & PRAM followed by Internet Recovery used to work occasionally, and when I used Disk Utility First Aid to check out the drives, they all passed with no issues, every time. Now it won't boot at all. Mostly it ends at the blank gray screen, but once I got a -1007f error message (which came up with nothing in a Google search).

I have separate clean installs of Sierra on each of the internal SSDs, and a Mavericks recovery partition, as well as a bootable flash drive with Sierra installed. None of them will boot. But I CAN connect in Target Disk mode from an iMac so it looks to me like the SSDs are fine.

Any recommendations? I am almost at the point of trying to re-attach the video GPU's heat sink, and/or bake the motherboard to reflow the solder joints. But I'd like to exhaust any software/firmware issues first that I may be unaware of.