What connotation does と思っている have as compared with the plain たく/ たがる form? I.e., how does this sentence differ as compared with 「首相はだれからも尊敬されたがっている。」？

@ericaw
～たがっている in this context sounds rude. You are mocking the PM.
～たがる means “eager to do X”. Therefore, it sometimes suggests that the person is bad at self-controlling or childish.
Now is March 2022. Russian President Putin is invading Ukraine. So, you may want to strongly condemn him by saying プーチン大統領はロシア国民から尊敬されたがっている。(= President Putin is egomaniac to be respected by Russian people). That’s the tone of ～たがっている.

～と思っている, on the hand, simply means “want to X” in a plain tone. It’s just objective and descriptive.