Why did the high temp thermostat in my dryer fail when I disconnected the buzzer?
As @brhans noted, you effectively made a short-circuit of the EOC buzzer. Buzzers can fail, and I suspect designed normally to fail open. If you wanted to exactly replicate the circuit, that would require putting in a resistor equivalent to the resistance of the buzzer. But most likely the dryer is designed so that buzzer failure will not prevent the rest of the dryer from operating properly, so removing the buzzer and capping the wires would have been sufficient.

Hopefully the Thermostat Hi-Limit Right was the only sacrificial component to this short circuit, but you may not know until you replace it.

My old Kenmore (really made by GE or Whirlpool, not sure which) has a knob to control the EOC buzzer sound - including "Off", but you're not so fortunate.

Adjusting sound level is probably not practical. But it would be easy enough to wire up an On/Off switch for the buzzer. However, it would need to be designed for 120V usage - i.e., like a full light-switch in a junction box with proper insulated wires, not a "little" switch despite it being "only a buzzer".