It seems that Saudi Arabia or Bahrain’s regimes tend not to get called 'illegitimate', so 'not democratic' would not imply 'illegitimate'.

I think it’s more like this: Governments the US doesn’t like very much are 'illegitimate' because 'not liberal democracies'; regimes the US does like are 'important allies' or otherwise not talked about.

The whole 'not democratic' thing is more like a type of post-hoc justification for what, to many, looks like blatant imperialism. It would appear it is successful at least some of the time.

Legitimacy is more about representing the people and ruling with consent. It's a spectrum, not a binary. The Saudi royal family have a patronage system in place that funnels money down to the regions and the tribes in return for loyalty. The country has no democratic traditions or expectations. For all they're a horrible dynasty of murderous autocrats, it's actually not that simple to argue that they don't rule with consent.

As governments get more oppressive and arbitrary in their exercise of power, it becomes harder for them to make credible claims of legitimacy. I'm not saying the Saudi government is legitimate, but it is possible for it to do things that increase or decrease it's relative legitimacy. Increase consultation and individual freedoms? More legitimate. Crack down on peaceful protests or oppress minorities? Less legitimate.

The strongest signal here is US ally vs adversary. Everything else is fluff in comparison.

Saudi Arabia does not get called illegitimate, but Cuba does.

Pinochet was 'legitimate'.