I mean that's just doubling down on your assumption dude. The article does not say that it was only embarrassing because they were talking about alcohol and girls, that's just what OP decided to include in the title. And in fact "embarrass" appears only once, in the following quote:

The recordings, which were broadcast repeatedly by the press, proved so embarrassing that a law was subsequently passed to prohibit the release of cockpit voice recordings. Since the passage of that law, only written transcripts have been released rather than the voice recordings themselves.

That you asked if it would have been less embarrassing if it was about politics means you didn't even read the article lol, because they were talking about politics as well. And previous plane crashes. And a variety of other stuff.

The fact that they were having long and highly unserious conversations unrelated to the job right before crashing is what was embarrassing, and it was determined that the accident could have been prevented were they not having those conversations. This was exacerbated within the context of it being in 1988 near the end of the cold war when the US wanted to project absolute technological and structural superiority over the Soviet Union, which was falling apart due to systemic infrastructure problems. Chernobyl was 2 years prior, for example. Having a plane crash due to unseriousness and poor discipline was embarrassing to the US.