The usual blood pressure measure instruments[1] work AFAIK just like the "old" manual way, what is measured is the (air) pressure in the band, though they use an oscillometric (tiny variations in the pressure) approach to detect the heart beats as opposed to the stethoscope (auscultatory).

The idea (the traditional way) is:

https://en.wikipedia.org/wiki/Sphygmomanometer

1) you place the stethoscope between the arm and the inflatable band

2) you pump air in the band

3) at a certain pressure you will start hearing the heart beats

4) you continue pumping until you cannot hear them anymore + a little bit more

5) you start slowly release pressure until you can hear the heart beats (at this point what you read on the manometer is the max pressure or systolic)

6) you continue slowly releasing pressure until you cannot hear the heart beats anymore (at this point what you read on the manometer is the min pressure or diastolic)

The electronic device replace the hand pump with an electric pump, the manometer with a pressure sensor and the stethoscope with the capability to detect oscillations in the pressure from the same sensor.

See:

https://en.wikipedia.org/wiki/Blood_pressure_measurement#Aus...

https://en.wikipedia.org/wiki/Blood_pressure_measurement#Osc...

I believe that the issue in a home made electronic device is not so much the sensor but rather the conversion from raw data of the sensor (oscillometric waveform) into heartbeats and the speed with which the band is inflated and deflated (size and speed of pump, accuracy of the release valve).

[1] both the ones that take pressure on the arm and those that take it at the wrist