Doge cuts spark questions as employees supporting Musk space launches spared

I would argue that this is more of an ambiguity caused by the English language, rather than a deliberate twisting of the headline. The word 'support' has at least two meanings - 'to advocate for' and 'to provide services for'. It's easy to mix them up when both are equally probable. But the real meaning you mentioned is evident in the article itself.

But even with that meaning, it still sounds like a conflict of interest. Musk retained the staff that he needed to sign-off on his space operations, while no such considerations were given to their colleagues. Further, those retained staff will feel the pressure to not offend Musk in any way, because he has demonstrated that they're disposable.

According to the article deep cuts were made everywhere except launch support staff.

So yes, that surely might raise proverbial eyebrows.