Why didn't ancient doctors use pulse as a measurement?

They did. The use of the pulse to check for life is recorded as early as The Epic of Gilgamesh. When Enkidu dies,

he raised not [his eyes, and] his heart, (when Gilgamish) felt (it), Made no beat.

The ancient Egyptian Ebers papyrus, from about 1500BC, discusses the circulatory system and the pulse (but, oddly to the modern reader, not blood - arteries were believed to carry air rather than blood), and there is some diagnostic use of the pulse. The ancient Egyptians have left us the oldest known record of quantitative measurement of the pulse rate, using a water clock to provide the timing.

While the diagnostic value of the pulse is limited, it is something about a patient that can be measured, and can at least often give a general idea about the health of the patient. For some diseases, it is a key diagnostic. Thus, it shouldn't be a surprise that the pulse was widely used by doctors. In ancient Chinese medicine, the pulse was considered to be a very important part of diagnosis; surviving medical works describing this date to about 300BC (and implausibly claim authorship by the legendary Yellow Emperor (Huangdi), 2600BC). It was also important in ancient Indian medicine, as early as 600BC.

In Europe, the ancient Greeks wrote on the pulse, and used it for diagnosis. Hippocrates, the "Father of Medicine" wrote about the pulse (but apparently did not use it for diagnosis). The ancient Greek medical literature credits Praxagoras of Kos, 340BC, for pioneering the use of the pulse for diagnosis. Herophilus, a student of Praxagoras, quantitatively measured pulse rates in the same manner as the Egyptians a millennium earlier, using a water clock, and he even built a portable water clock, which he could take to the patient. The ancient Greeks understood that there was a difference between veins and arteries, and recognised that veins carried blood. Like the Egyptians, the ancient Greeks believed that arteries carried air, and probably for the same reason (probably due to the fact that on death, the arteries contract, and if the arteries are then opened, they are found empty of blood, while the veins still contain blood after death). Notably, our word "artery" comes from the ancient Greek αρτηρία (arteria), which can be translated as "air-containing".

The pulse continued to be used for diagnostic purposes through the Middle Ages, which, of course, continues in modern medicine.