An engine system is provided that includes a gas turbine engine and an actuation system. The gas turbine engine includes a compressor section, a combustor section, a turbine section and a flowpath extending through the compressor section, the combustor section and the turbine section. The actuation system includes a fluid reservoir, a flow regulator and a flow circuit. The flow regulator is configured as or otherwise includes a barrier between the fluid reservoir and the flow circuit. The flow regulator is configured to fluidly decouple the fluid reservoir from the flow circuit when the barrier is intact. The flow regulator is configured to fluidly couple the fluid reservoir with the flow circuit when the barrier breaks. The flow circuit is configured to direct gas from the fluid reservoir into the flowpath when the barrier breaks.