I have been going a bit back and forth on this question because the material on the Luftwaffe's manpower recruitment, especially in the early war years, is relatively thin. The destruction of a lot of Luftwaffe records in 1945 did not help either. But as a whole, the family stories and hypotheses are not very plausible.

Conscription within the Luftwaffe was a complicated animal in the peacetime and early war years. German males completing their year of compulsory labor service (RAD) were then drafted in their respective Wehrkreise. Upon completion of the RAD, German men could volunteer for either an extended enlistment in the Heer or volunteer into service in Kriegsmarine or Luftwaffe (the Waffen-SS comes into play during the war). In the later cases, volunteering also had the benefit of choosing which branch of the specific service (submarines, pilot training, etc.) the volunteer wanted, although in a pattern familiar in many modern militaries, volunteering for a specific branch within the Wehrmacht was not a guarantee the applicant would get it. The result of this was the early war Luftwaffe's personnel had a strong volunteerist character to it, albeit the volunteers were operating in a dictatorship that allowed next to no opportunities for conscientious objection.

But the available sources suggest that, particularly for the early years, the Luftwaffe had a strong cohort of volunteers within it. Flying and the allure of high technology drove a number of Germans to want to join up in this service. The allure of clean beds at night, less formal military discipline, as well as technical education also held an appeal for many Germans going into this branch of the service. Pilots and aircrew in particular were by and large volunteers. Training for these positions took years, although wartime pressures cut that down considerably. The Luftwaffe leadership was quite careful in trying to screen out applicants to these posts.

Now there were exceptions to this general outline. Germans with a civilian pilot license were inducted into the Luftwaffe once hostilities began in 1939. This was how the postwar sex entrepreneur Beate Uhse and Melitta Schenk Gräfin von Stauffenberg, sister-in-law to the 20 July plotter, found themselves in Luftwaffe service despite being women. However, pilot licensing in Germany was not a politically-neutral affair. Both the Deutscher Luftsportverband (German Air Sports Association) and its successor, the Nationalsozialistisches Fliegerkorps (National Socialist Flyers Corps) had enormous ideological and financial support from the dictatorship. Pilots emerging from these groups were exposed to a considerable amount of indoctrination. Although Luftwaffe memoirists often played down ideological within their prewar civilian flying education, Hitler did reportedly quip he had a monarchist navy, a reactionary army, and a National Socialist Luftwaffe. The Luftwaffe also did absorb the bulk of the Austrian armed forces during the Anschluß, including the comparatively small Austrian air force.

So it is possible that the ancestor in question was a drafted pilot, but it is unlikely he was compelled to serve as one, especially in the early war years. The lengthy screening process for pilots such as vision tests made it very easy to wash out. Nonetheless, flying for the Luftwaffe did not necessarily mean being a true National Socialist. The Wehrmacht, including the Luftwaffe, did have its share of grumblers and unenthusiastic soldiers. The war itself did not really become popular in Germany as a whole until the massive victories of 1940. Internal SD reports often highlighted this general lack of enthusiasm until the spring of 1940, even if such flagging spirit did not translate into open opposition. The internal security apparatus of the dictatorship, which was often quite small compared for the task of policing the whole of Germany, reacted to petty grumbling with guarded indifference; Germans denounced to the Gestapo usually received a stiff warning and further monitoring rather than further punishment if the accused did not translate grumbling into open acts of opposition.

Popular memory though holds that the SS terrorized the whole population into reluctant obedience. This was very prevalent in the immediate postwar years and has continued in both popular media representations of the Third Reich as well as familial stories about life under Hitler. Historians such as Eric Johnson, Peter Fritzsche, Nicholas Stargardt, Michael Wildt, as well as studies like the University of Hannover's Opa war kein Nazi have demonstrated less terror and more cooperation with the dictatorship among the wider German populace. In the Opa war kein Nazi project, the team discovered familial narratives often projected resistance and victimization upon parents and grandparents that did not exist.

State violence and threats of violence did exist though. The Third Reich did employ the Sippenhaft, family collective punishment, for those who committed treason. But as Robert Loeffel in his study of family punishment showed, the dictatorship' application of Sippenhaft was often patchy and up to 1943 largely limited to outright political opponents of the regime. Families punished for actions of one member do go up after the Stalingrad defeat and escalated after the 20 July Plot, but even here there were limits. There usually needed to be a more visible form of treason, such as joining the Soviet-sponsored Nationalkomitee Freies Deutschland. Yet such terror could still be selective. Even the aforementioned Melitta Schenk Gräfin von Stauffenberg was able to resume her flying duties at Rechlin while her husband was imprisoned at Buchenwald (tragically, and ironically, her husband survived and she did not as she took off in a plane during the last days of the war flew to the prison she thought he had been transferred to and an American fighter shot her down).

But all the above is only a general outline of what the OP's great-grandfather did during the war. A lot of research has shown that family memory mythologizes and invents resistance, but that does not mean such stories are automatically false. However, the odds are against it.

Yes. I have previously found answers by /u/kieslowskifan and /u/vonadler and /u/commiespaceinvader for the broader question How did conscription work in Nazi Germany?