In ancient times, taking the pulse of a patient was a common way to determine if there was any sickness in the body. There's a interesting research 1 where it writes (Hajar R., 2018):
Ancient civilizations linked the heart and heart or pulse rate to life. We know that the Egyptians and Chinese of antiquity were masters in pulse diagnosis and Chinese traditional medicine still emphasizes such art. The ancient Egyptians believed that the heart gives rise to vessels that lead to different parts of the body and that its motion can be felt at different peripheral sites.
From ancient times, the pulse has been recognized as the most fundamental sign of life. If somebody collapses on the street, a bystander automatically feels for the victim's pulse. Early physicians paid great attention to the character of the pulse in health and disease. For thousands of years, books on the pulse and how important it is have been written, with Galen the most prolific and who has written volumes on the pulse. Many of his descriptions are still being used today.
In the 18th century (1707) with the invention of the “Physician's Pulse Watch” (a watch with a second hand that could be stopped), changes in pulse rate in health and disease could be accurately assessed. The electrocardiogram (ECG) was invented in 1895 so that analysis of the pulse’ contours advanced further. In the 1960s, digital signal processing techniques proliferated so that now physicians have only to look at the screen display by the patient's bedside to assess a patient's clinical condition. Many doctors feel that feeling the pulse of a patient is no longer necessary and many physicians ignore the arterial pressure tracings displayed on cathode ray tubes in intensive care unit and consider only the numeric display of blood pressure worthy of attention. However, examination of the arterial pressure contour is a clinical tool capable of sometimes providing important information on cardiovascular function. Hence, besides the pulse rate, its regularity and volume are noted.
In halacha however, there's a debate whether to follow the ability of breathing or whether we feel a pulse. (See Admur 329:3 [follow breath]; Michaber 329:4; Yuma Mishneh 83a; Gemara 84a for a Machlokes if we follow the heart/pulse or the breath).
Rav Moshe Feinstein is quoted in saying that "a victim may not be breathing because of temporary nerve damage". Maybe this is why the pulse was tested.
1 Hajar R. (2018). The Pulse from Ancient to Modern Medicine: Part 3. Heart views : the official journal of the Gulf Heart Association, 19(3), 117–120. https://doi.org/10.4103/HEARTVIEWS.HEARTVIEWS_16_19