Our thoughts are peopled. Listen to your thoughts. When we say thoughts ride on a sea of feeling, that often means they’re entwined with an interlocutor or an audience – other people, that is. When we engage with other people, in our heads or in the world, emotion is part of the equation. When we’re talking in our heads, we’re talking to someone, a real person or some vague other.
That someone could also be ourselves. As with all communicative acts, we are seeking to influence an agent - another self within the multitudes. When we're talking to ourselves (silently or aloud), we're engaging in a goal-directed behavior, such as trying to strengthen our resolve (yes, I can!) or remember something (broccoli). We’re still riding on a wave of feeling, because goals can't gain traction without emotion. For instance, vigilance fuels self-talk in the service of resolve or memory and you can’t be vigilant without a bit of anxiety. Anxiety is a low-status emotion these days, so we may call it something else, like "concern".
Of course, not all “task-independent” mental activity involves words or goals. Our heads may be filled with visions of bread pudding.
Then again, a vision of splendor could very well motivate goal-related thoughts on how to achieve such a wondrous thing.