” whether Yiddish is (still) a true Jewish language “
As long as there are gedolim who still speak it, I’d say its still a language that is distinctly Yiddish. And by the same token, English, when used to convey a Torah concept, is also (in a manner of speaking) a “Yiddish” language. This was the approach of R’ Yaakov Culi, in his M’eam Loez, so I would imagine it applies to the language spoken today, whatever it may be.
As far as feeling uncomfortable in the presence of a conversation when the language is one not understood by all present, that’s how I feel when I hear fluent Ivrit. If they want to include me, we either switch to Yiddish (Yerushalmis speak it) or English (most everyone else speaks that)
For the handful of folks that speak neither, we struggle along in my loshon koidesh / piecemeal Ivrit.
The point is, communication. Sometimes its easy, sometimes not as easy. But we get the job done, because we have a common objective.
Of course, I don’t have the same opinion about the Yiddish spoken in the Yiddish Theater, or by “Yiddish-ists”