Replacing a 12v battery (model A23) with power pack
Older transformer based power supplies only have the stated voltage close to the stated current. If you measure the 12v 2 Amp or the 0.6 A supply with nothing connected, is it still 12V? Also, even newer switching supplies have a minimum current for stable operation (Typically 10%). That means the 2A supply might need atleast 200mA draw to have a regulated 12v.

The 23A is essentially 8 coin cell batteries in series. It is 55maH capacity, and it's internal resistance is unknown. This internal resistance acts like a resistor in series with any connected load, so current would be limited. This is why you can connect an typical LED to a 3v coin cell without a resistor, and not blow the LED. It simply can't provide too much current without it's voltage sagging (voltage droop). A generic datasheet for a generic 23A shows that with a 100Ω load (for 0.3 seconds testing), the battery voltage drops down ~4v from 12v to 8V. That's only 12v / 100 =, or 120mA draw.

Basically, what happens is that the battery itself acts like a resistor, preventing high current draw and dropping it's voltage as current draw increases. A remote could have been designed around this fact, as a way to protect the remote. Or your power supply is a unregulated/badly regulated supply. Since the caps blew, this might be the case. Caps normally blow because of high voltage. Test them first, if the 12v 600 mA is 12v without a load, try connecting it to a remote with a 100Ω resistor in series. This should limit it somewhere between 120mA to 10mA draw (depending on how much voltage the resistor drops).