Perhaps aides think somehow they are being supportive, but if I were that powerful person, I believe I would rather just a few people I trust to tell me when I'm effing up - than everyone telling me how fantastic I am.
The majority (by far!) of my colleagues are male, and part of the subtext and context of this type of conversation is (1) to comment on things guys "can't" comment on, like "did I wear the appropriate jewelry or footwear for this event to project the social and intellectual persona that I'm trying to project", and (2) to combat the everyday little degradations you encounter, like someone at the registration table not registering you because they figured you're a girlfriend not a real participant. Women who are trying to maintain and accumulate power know that they can support each other in maintaining a stance of confidence rather than acceptance. Being primarily socialized into a stance of acceptance due to my historically all-male and socially inept professional circles, I was way confused when I got a female boss who started saying nice things about/to me. It is true that I do good things at work, although I don't manage to find the right shoes to project being corporate -- but I've never had anyone say anything about either of those things! Ever!
It's a reasonable management tool in some ways. My male bosses asked me (forced me?) to do my job and also take care of "women things" (recruitment and retention of women and minorities) without ever saying anything nice or giving me any credit, so I quit doing most of that. Altruism has limits that are often reached at the bank account. But now I have a boss who views outreach as a positive and actually says nice things ("you're incredible! I'm amazed at how much you do!" etc) so yep, I'm more likely to do it. And it is good for my organization.
Maybe I'm weird, but external validation doesn't really push any of my buttons. If you think I really did a great job, give me a bonus, or an extra day or two of vacation. Otherwise, I've got work to do.
Edit: I guess for context I should add we're both male.
Now I'm working with some folks from a more corporate background, and it is a different culture. Very different. I'm still getting used to it myself.
Doing a good job != standing out
"Recognise and reward" should be a tenet of good management, and you don't reward people for doing what they should be doing anyway.
One good reward is seeing if there are any conferences in the person's specialist field. But so they know it's not a holiday, have them to a brief presentation about what they learned upon their return.
The amount of effort required to be sycophantic is minor in relation to the past 10+ years of work that your superior could torpedo on a whim.
Don't get me wrong, it's still a hierarchical, traditional organization.
And if you were powerful, and wanted to, you would likely be able to find ways to surround yourself with such people. But I think often enough, the people who want power don't want it because they're very well-centered, and I think it's fair to assume that politicians and other people mostly select the nature (which can mean what those really think and/or what they claim they think) of their "underlings" by the signals they send, the incentives they give, by whom they hire or fire. In other words, the dog is probably wagging the tail, not the other way around.
I could be wrong, I don't know this woman and don't follow US politics that closely, but just from random things like this https://www.youtube.com/watch?v=-dY77j6uBHI I would call it a somewhat educated guess. Some people don't want to know the truth about themselves, that's the exact opposite of their mission in life. By which I don't mean to bash Clinton in particular, the list of big time politicians I wouldn't suspect this about would be very short and hard to come up with.