Same issue with my early-2011 13" MBP. THE OEM battery exploded and almost took out the trackpad and case. I got it out in time and took to Apple. They recommended I buy a replacement off the net. I did.
I took the opportunity to also upgrade to a SSD drive.
Now my MBP won't turn on unless I remove both the power cord and the battery. If I do that and put the battery and power cord back in place it will start every time. The next time I shutdown I have to yank the battery and power cord again.
We tried SMC reset. It did nothing.
I hope someone figures this out eventually. My laptop is fine other than this annoying issue and Apple says it's now "vintage" and won't work on it.