Here is my crack at it. I don't know many advanced circuit designs, but a rudimentary design might look like this.
Sensor
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Array of bandpass filters (your monitor's accuracy will depend on the filter count and resolution)
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Each bandpass output feeds to a smoothing filter to get a measurable amplitude (voltage) for each freq range
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Logic circuit which compares the filter outputs (voltage) and identifies the strongest voltage (aka the freq range which the heart rate falls into).
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Output of logic circuit is a quantized voltage specific to the heart rate measured. An ADC will "decode" this output voltage giving you the bits you need to drive the display.
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The ADC output is run through gate logic to convert to display segment bits.
This avoids programmable controller chips, but after thinking it through, it seems a bit convoluted, no?
Unless this is just a thought experiment in "how could it be done without a ?C", I would say this is exactly why microcontrollers were created, and excluding it from design is like building a skyscraper without steel.