There may have been obstruction. In the instance given , depending on how far the ball travelled past the second baseman, the position of the other fielders, the degree to which the collision hindered or impeded the runner, and possibly the subsequent actions of the runner, the umpire might award the runner third if in his judgment the runner could have reached. Had the runner continued to third and been thrown out, the umpire would almost have to award third unless the collision was minor and did not hinder or impede. Contact by itself does not constitute obstruction, but it is an awfully hard sell not to call it, especially if the runner is tagged. Many times I've seen clever batter-runners fool a novice umpire into an extra base by rounding first and contacting the first baseman with subsequent theatrics and encouragement from the first base coach.
Obstruction is the act of hindering or impeding.
Contact alone does not constitue obstruction, although in most instances it does, reasonbly, entail hindrance. Obstruction is always a delayed dead ball situation. To rule otherwise penalizes the offended team. The umpire's signal is not mandatory, but as you indicate, the failure to call it constitues poor umpiring. But his failure doesn't negate the fact, and the offended team should not be penalized for it.