Why didn’t you try the battery and took the nuclear option first? It’s often caused by a battery issue - on my E6540, my machine didn’t boot the first time and I either had to wait or press the power button twice to get a boot. When I ran it without the battery, that completely resolved the issue. The battery is more notorious for creating boot problems.
In your case, the board was not likely to fix it and it was an expensive gamble. When it comes to the E7250, I see a lot of those with bad board like symptoms sold for parts.
One or two isn’t concerning, but when you see them in large numbers there’s a systemic hardware issue. The Intel HD/Iris Latitudes are bulletproof - it’s the ones with a GPU that are more known to have hardware flaws like dying GPUs - the E7240 is too small to even have room to introduce that failure point. In general, a whole new board is about as much as a No HD machine you can transfer the ownership on and swap drives over, so that repair is typically not economically viable unless you get the board dirt cheap once it’s a confirmed board issue.
Since you already tried the board, try disconnecting the keyboard and trackpad, and check for a boot. It’s kind of a long shot, but seeing as you changed the board that’s the last major part I could think of that would cause it to not boot and there’s not much left to try.
Did you repaste the CPU? That will do it too.