Actually, a third strike and an infield fly are treated in a similar manner. The general rule is that the batter becomes a batter-runner if he hits a fair ball. Perhaps it isn't obvious, but the general rule is that he also becomes a B/R on any 3rd strike not caught. Go back into the history of baseball far enough, and the rule was that he became a B/R on any 3rd strike. In fact, in the the NFHS formulation of the rules, even today the batter becomes a B/R on every 3rd strike, but is out immediately if the pitch is legally caught (or 1st is occupied with less than two out, of course).
Both situations offer the possibility of a double play if the fielder or catcher deliberately misplays, either by letting a batted ball drop untouched, or by not catching the 3rd strike. The rules today cover both situations by making an exception to the general rule: The batter is out if a double play is a likely outcome.
For an infield fly, it is necessary for the umpire to make a declaration, since some judgement is involved in determining if the batted ball is in fact an infield fly, and the correct action by the runners depends on that declaration.
For a dropped third strike, where a double play is possible, no judgement is required. The batter is out on strike 3 whether the ball is caught or not. So the umpire does not need to declare the out on the batter, and both the offense and defense should know that the batter is out. The situation was evident before the pitch, the out is automatic, and the runner's actions should be the same whether the pitch is caught or not.