Why would a COB be utilized in this circuit?
Was there a resistor under this paste?


No.

The "paste" was an epoxy coating that was protecting a "chip on board" IC.

Most of us are familiar with ICs in rectangular black plastic packages with metal legs. There's an unseen tiny sliver of silicon embedded in the centre of those. However, for very high-volume manufacturing, it is sometimes cheaper to dispense with the plastic and metal leg packaging and instead attach the tiny silicon IC directly to the PCB.


  I expected a tiny circuit with a button and resistor.


What you got was more sophisticated than that.


  how did the resistance work in this circuit?


Either the IC was regulating the current or (more likely) the circuit design relied on the internal resistance of the battery. For example, a 3V lithium coin cell has about the right resistance to make a small-output LED work safely - see LED throwies