The Bass Can Remember a Different Song
The lowest voice can preserve a history the chord above it might prefer to forget.
Hold C, E and G in the upper voices. Over a low C, they give a firm sense of C major. Over A, they complete an A minor seventh. An analyst can give each sonority a name, and the name changes. A listener can still be hearing much the same colour in the voices they were attending to. The bass has changed the conditions in which those notes are understood before the person singing one of them has had to move.
That is one of the reasons an arrangement can alter an emotional relationship without asking a singer to alter a performance. The singer repeats a phrase with familiar emphasis. A shift underneath it lets a sound that previously felt like rest acquire a small question. The effect is especially persuasive when the bass appears to be continuing a line the listener already had reason to hear. If it simply lands on a clever note, an audience may experience a local effect. If the note is the consequence of a shape taking several bars to form, the new chord arrives with the weight of an implication.
More than the root
One way to hear an inversion is as a better or worse fit for a chord function we have already decided. Another is to give the bass enough independence to make a different decision plausible. A bass voice that goes down through the change may preserve the direction of a phrase even while the upper notes have started to turn toward a new section. The music acquires two accounts of where it might be headed. Sometimes the first account is vivid enough that a chord a theorist would call stable still sounds unsettled.
This effect can be oversold. A complicated bass under a clear melody can make every moment seem to be about ingenuity in the lower voice. An effective change may be the bass holding still while the rest of the music passes it. There is room for other experiences when the player knows which motion is worth calling attention to.
A useful analysis can ask what the top of the arrangement believes is happening and what the lowest voice can remember. They might agree for an entire verse. A dramatic moment can arrive when the bass will no longer support the explanation the melody has been giving. A listener does not need to name any of the chords to know that the singer seems to be telling the same story from less secure ground.