"blood-letting," c. 1400, flebotomye, fleobotomie, from Old French flebotomie (13c., Modern French phlébotomie) and directly from Medieval Latin phlebotomia, from Greek phlebotomia "blood-letting," from phlebotomos "opening veins," from phleps (genitive phlebos) "a vein" (a word of uncertain origin) + tomē "a cutting" (from PIE root *tem- "to cut").
word-forming element meaning "one who does or makes," also used to indicate adherence to a certain doctrine or custom, from French -iste and directly from Latin -ista (source also of Spanish, Portuguese, Italian -ista), from Greek agent-noun ending -istes, which is from -is-, ending of the stem of verbs in -izein, + agential suffix -tes.
Variant -ister (as in chorister, barrister) is from Old French -istre, on false analogy of ministre. Variant -ista is from Spanish, popularized in American English 1970s by names of Latin-American revolutionary movements.
<a href="https://www.etymonline.com/word/phlebotomist">Etymology of phlebotomist by etymonline</a>
Harper, D. (n.d.). Etymology of phlebotomist. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/phlebotomist