Yes, they should but they don’t because undertrained techs let time pressure and faux efficiency overtake best practices….over and over and over again.
I’ve had this experience at far too many places too. There’s one dr’s office I go to routinely every 3 months that I just stoutly refuse to let them use their excruciating automated BP machine on me anymore (I gave it a chance but after the third consecutive visit that left me in tears before I got to the 31 injections, I now just say no firmly). I always offer that they may take my BP the old fashioned way with a cuff, a stethoscope and their ears but many of them have never been taught that prime diagnostic skill. Grr….