MacBook Pro 2011 has faulty power button - How do I enable any key or lid open boot settings?

Opening the lid to turn on a Mac only works if the Mac is asleep/hibernating. If the power is off, it will need to be turned on.

The reason you have to disconnect the battery is because although your power settings may call for "boot after power failure," there's never really a power failure if you're battery is connected (and has sufficient charge).

There's two ways you can address this:

• Short the power pads. This is a "stop gap" way to turn on a Mac so that you don't have to disconnect the battery. Simply bridge the two pads with the blade of a slotted screwdriver or, if you're really daring, solder a wire loop (long enough to come out of the case) with a momentary switch to make it more convenient. This way, you don't have to keep opening the back cover. It's a total bodge, but it'll get you by.

  

• Replace the keyboard. I actually reference this exact keyboard in this question/answer: Keyboard key stuck or not being recognized; how to fix. Look under "Replacing Keyboard" in the Answer for specifics. It's actually very inexpensive. You may also be able to find the button by itself on eBay.

I advise the latter. You may be able to get away with only replacing the button rather than the whole keyboard, but either way, it's a better, more reliable fix.