Although probably rare for the rather abbreviated "induction" physicals, erections were quite common during the "cattle drive" styled pre-induction physical exams which were nearly a universal experience for all American 18 to 22 y/o males most recently during the Vietnam War draft. Following mandatory Selective Service registration upon reaching the 18th birthday, whether ordered to the examinations by a draft board, or seeking enlistment into any branch of the military service, it is hard to forget spending most of the day totally naked, or wearing briefs going station to station in groups ranging from several hundred to over a thousand young men.
As a 91-Bravo medic I not only went through the experience (twice) but I was also "privileged" to spend some of my Stateside duty time assigned to assist at two MEEPS examination centers.
Yes, erections happened. Some guys no doubt got a little excited watching others getting the hernia and external plumbing examined. Others might have found slightly more than passive interest in seeing 200 or 300 schlongs donating their piss into the glass specimen bottles, and others, myself included experienced my dick getting overly happy performing the nude exercises. Some no doubt got erections only because of testosterone overdrive at their ages. For whatever reason, it wasn't unusual to observe a firmly tented pair of tighty whities, or a dick or two of varying length pointing at similar angle as an artillery cannon.
Certainly unprofessional, but some of the medics, corpsmen and other NCO's who got assigned to the monotony working at these examining centers, found it entertaining to do everything possible to embarrass and humiliate the examinees who had undisciplined "short arms". I am confident that no one was ever disqualified from conscription because his dick came to involuntary attention.
I spent nine months at a basic training military base in Missouri, and did more than my fair share assisting with reception physical exams. Smaller groups, with only about 100 to 150 examined each day, but still far more men than the WWII era building was designed to handle. No briefs or boxers.
Nudity was the uniform of the day, and yea some guys seemed so excited about their new employment in the Army that they also presented serious hard-ons. No big deal. If the medical officer noticed the erection he would simply walk up to the recruit and politely say "at ease, soldier"