As for single words for the phenomenon you describe:
Each of the above words can have a negative connotation, particularly when the person who is being circumlocutory or periphrastic is attempting to evade having to talk about something, or when the person does not know what he or she is talking about but feels the need to fill the silence.
For example, a person might evade having to answer the question, "What role did you play in your employer's attempt to swindle people through an elaborate Ponzi scheme?" by saying,
I'm glad you asked that question. So many people avoid asking the hard questions, so personally I commend you for the forthrightness of your question and for giving me the opportunity to respond. Moreover, I want to respond to your question, because there are probably many other people who have the same question you posed to me but were afraid to do so. I commend you, again, for your fearlessness. You've reminded me just now of an incident which occurred to me a number of years ago. Was it 1998? Or 1999? I'm not exactly sure, but the incident involved a person such as yourself who asked me a pointed question regarding a situation in which I was tangentially involved . . ..
A less pejorative way of describing the phenomenon is to use the word obliquely, as in the following:
- Alice approached obliquely the question she really wanted to ask of Bob.
Or, the word diplomatically communicates a great deal when combined with other words, particularly the word skirted. For example,
Alice wanted to ask Bob a question diplomatically, so she skirted the issue deftly until the time was right and she felt comfortable asking the question.
Again, to "skirt" an issue can involve a measure of avoidance or evasion and, consequently, some periphrasis or circumlocution, but not necessarily.
The expression "to beat around the bush" can involve empty blathering, but not necessarily. Beating around the bush can involve diplomacy, particularly when a subject or question is difficult to broach head-on. In that case, the questioner engages in a little good-natured bantering and only when the time is right will pose the question.
ADDENDUM, in response to revision of question
The technique utilized by Carol is a pretty basic rhetorical strategy which goes back to ancient Greece, where public speakers would establish their ethos in various ways throughout the speech. Aristotle suggested that ethos (modern day, "credibility") has its source in a speaker's perceived intelligence, virtue, and goodwill. Sounds to me as though Carol is establishing her credibility by enhancing the goodwill between her and her audience by reminiscing about her childhood friend.
Audience members can obviously identify with her reminiscences and may even be moved emotionally, which is evoked by Carol's use of pathos (Gk., for emotional appeals).