I'm fairly sure Courant was persecuted for being Jewish. Other biographies are pretty clear on this and I can find no reference to what political party Courant belonged to in Germany.
Courant left Germany in 1933, earlier than many Jewish escapees. He did not lose his position due to being Jewish, as his previous service as a front-line soldier exempted him; however, his public membership in the social-democratic left was reason enough (for the Nazis) for dismissal
I'm sure Courant was persecuted for being Jewish (see e.g. the discussion in that same document about how Landau was treated), but it seems to have not been the official reason for his dismissal.
[1] http://irma.math.unistra.fr/~schappa/NSch/Publications_files...
For example, the Nuremberg Laws (which, among other things, banned marriages and extramarital intercourse between Jews and Germans) only went into effect in 1935.
Considering the Nazis only were in positions of power for twelve years and still managed to murder six million Jews and many million other people they did not like, they did move quickly and there certainly were sudden shifts and changes in what people (in particular Jews) were allowed to do and subjected to. However, the overall trajectory was still one of turning up the intensity, not going into it with full intensity, among other things certainly also for propaganda reasons.
Given that the Social Democratic Party was the only part voting against the Enabling Act in 1933 (Communist MPs were at that point already all imprisoned, banned from parliament or fleeing) he certainly had, however, two things the Nazis could attack him for, whichever was more convenient.
So makes sense to me that they pushed him out and makes sense to me that he fled as quickly as possible.
The Nazis murdered plenty of Social Democrats. The Nazis obviously murdered plenty of Jews. If someone was Jewish and a Social Democrat the distinction of why the Nazis were persecuting them becomes sort of an academic one. They might name either or both reasons, whichever sounds more convenient.
That's a bit like saying nobody in the US gets shot by police because they're black, since that's illegal. Anti-semites in Germany didn't suddenly start liking Jews who had rights.
(I left off reading Weil around this point. I'd get back to it, but find political systems without a healthy number of major parties rather depressing subjects.)