You do not need acute lead toxicity to cause harm and hunting certainly is a source of lead exposure in humans.
"The highest single measured blood lead concentration in this study was 221 mg/L. This is 3e4 times lower than the level that could be expected to cause lead poisoning, but effects on the hematological system, hypertension and renal function is a possibility among the participants with the highest blood lead concentrations. However, the most critical risk of lead in humans appears to be effects on the central nervous system of fetuses and children. Blood lead concentrations between 50 and 100 mg/L, but also as low as 30 mg/L may have an impact."
https://www.biologicaldiversity.org/campaigns/get_the_lead_out/pdfs/health/Johansen_et_al_2005.pdf