The main problem with move counters is that they tend to be wrong because of players hitting the clock too many or too few times.
A second problem is that with most digital clocks it is not possible to see EXACTLY how the clock is set without fiddling with it. Considering all the stuff I can see on my digital wristwatch, the failure of chess clock designers to have a display big enough to show the full settings is frustrating.
Perhaps what is needed is a THIRD button somewhere that can be pressed after both players and/or the arbiter agree that the time control has been met which then moves to the next time control setting.
Personally, as a TD, I don’t hate the move counter. In fact, I recognize that since some players like it, it should be the TD’s job to deal with it when necessary.
Some TD’s, however, don’t like to keep an eye on games to make sure everything is running smoothly, and I can’t really blame them. If a move counter error occurs near the time control, or occurs earlier but is not noticed until near the time control, there can be some unpleasant consequences.
For example, if 40 moves have been played but the clock shows 39, one or both players might worry about still being in time trouble and rush their moves unnecessarily. Or, if only 39 moves have been played but the clock shows 40, a player might be lulled into relaxing, only to have a time forfeit claimed by his opponent (and upheld by the TD).
When I, as a player, use the move counter, I try to watch it like a hawk. If the opponent forgets to press his clock, I’ll remind him. After he forgets for about the third time and I don’t feel like reminding him anymore, I’ll just wait till I move, then press first his clock and then mine, to correct the move counter.
So, despite how nice the move counter is (in my opinion and yours), there are some legitimate pitfalls.