... don't forget that distribution is a ... well, not a bell curve, but a pyramid. So, SOC2 anybodies will be few and far between, as will the SOCC anybody. Unless your players are playing with loaded dice, you won't get many SOC2 doctors anyway. Solving them on an individual basis works just fine.
I wouldn't call 1/36 "few and far between." Unusual, but not so unlikely that I'd question it - party of 6 has about a 15% chance of having someone with a SOC2, or a SOC 12 for that matter. 1/216, that'd be "few and far between."
...There are no impediments getting into the Doctor's career from having low Soc. More to the point, even if there had been, once someone from the gutters has gotten into college and through medical school, his Soc would almost certainly rise to equal his new position in life. ...
Kinda depends on your society and your reason for having a low SOC. Consider that black Americans pre-1964 had a very low social standing, and becoming a black Doctor didn't do much to change that in many parts of the country.
Which brings up a different sort of problem: yes, what constitutes high society is pretty well universal in Traveller culture, but what constitutes low society is likely to vary quite a bit from culture to culture. A Dalit "untouchable" is only an "untouchable" in old India and related Southeast Asian cultures; get away from the culture, and he's just an ordinary Joe - and with a bit of luck can even aspire to high status. In a similar vein, the black Doctor was low status in the dominant culture, but he enjoyed high status within his own subculture.
The SOC stat is unusual among the various character stats for trying to capture a feature that depends more on how others see the character and less on anything intrinsic to the character himself. What constitutes "low brow" or "peasant" garb or behavior varies widely from culture to culture - and with changing styles. An intelligent person can learn to adopt the behaviors of a higher or lower class, in essence raising or lowering his apparent SOC and affecting how others interact with him (though failing in that effort - in either direction - may have some very severe consequences).