"All this talk about steps not landing on the beats (or syncopated beats, whatever) leaves me wondering how you stay with the tempo of the music. I don't know if it's a matter of symantics or what."
It is basically a matter of semantics. In a dance like waltz, most of the steps probably do land on the beat for some reaosnable definition of "land" and "beat".
However, when we formally speak of a "step" in dancing, we are speaking of a collection of actions that takes place over a period of time. At some point, someone maybe a decision and put the dividing line between these steps at the point where the feet pass, rather than at the point where the foot is placed or weighted.
In other words, steps - and the actions they are required to include - begin some noticeable time before they land, and continue some noticeable time after they land. If the landing is to be on the beat, then the time period of the overall collection of actions assigned to that step beings before the beat, and ends after it. Of course you only need to get into that if you want to debate which step period an action is assigned to - or if you want to consider which beat an action belongs to, you have to take into account that the overal periods of the steps defined this way are offset from the beats - the beats land near their middle, not at their beginning.
Of course you could also ask where the instant of a beat is, and how that might correspond to motion. For a drummer, the sound comes at the end of the motion (really where the downstroke becomes a rebounding up). For a violinist, the sound occurs during the stroke of the bow. What about for the conductor? I think it's near the bottom of the baton's movement... but maybe it depends on if the conductor originally trained on violin or percussion? Fact is, the orchestra quickly adapts... as most likely do dancers. People may feel that "on time" is an objective measure, but it can actually be demonstrated to be quite subjective.