There is no conflict. The team member who is on the bench in your Case Play has not been DQ'd, but that doesn't automatically make him an eligible substitute. In this case, a specific substitution rule forbids his entry at that point in the contest and he is therefore an ineligible substitute. Thus the team has to continue with four until he becomes eligible. That isn't a difficult concept. You just don't like it, so you conned the NFHS into overriding a rule in order to do what you desire--allow a team member to return to the game improperly.

As for the TF situations prior to the start of the game, the RULES contain specific exceptions to allow for substitutions in those cases.

Is there any medic on the scene that can determine whether an injury is deemed serious that the player should not be allowed to continue to play?

If there isn't, then it's up to the coach & player to determine whether if a player is fit to play and the ref should decide whether to allow the player to continue.

I mean if the player rolled the ankle but can still stand and move around, it would still be better to put the player at the defensive side so while the team is facing 4 on 5 on the offense, they'd still have 5 players on the defense.

The "ref should decide whether to allow the player to continue" is not something I've heard (outside of maybe some youth rec ball leagues). For scholastic/HS games, we can remove a player for exibiting symptoms consistent with a concussion or who is bleeding or has blood on uniform. And we can stop play and beckon a coach/trainer for any other injury and that player must be replaced or a TO taken.

Beyond that, if a player reports into the game, I'm assuming that he/she is cleared by a trainer and is fit to play. Schools around here are pretty good -- trainer is always around to take a look at any player who leaves due to injury.