Hi Josh,
I had a similar issue with a Rane SL2 interface; I could only power it up by plugging in the adapter to wall power. It turns out that the capacitors in the wall wart (a switching-type power supply) had leaked over time and were now "marginal", but seemed to receive enough of a 'boost' when initially plugged in to overcome the unit's initial current draw. I tried another adapter and the device worked perfectly; the wall wart was fine after replacing the capacitors (with higher-temp spec ones).
All of that to say: have you tried another compatible 15V 2A positive tip adapter?
Hope that's the issue; 'momentary' power switches (as opposed to physical ones) have their share of idiosyncrasies - if it's not a firmware or wall wart issue, it may be a more complex problem.
Good luck!
Andrew