Under NFHS (Federation) rules, if a player on the bench (not a field player at the time) receives a straight red card (not taunting, or excessive celebration or a second yellow)...that player is disqualified. That is explicit in Play Rulings page 65 12.8.3

My assumption is that the team does not have to play short.
Correct?

On page 39 Rule 12 Sec 8 Article 5 says "A disqualified player cannot be replaced except as in Article 2"
Article two says "...number of player shall not be reduced for: taunting, excessive celebration, 2nd caution"
This is for player, coach or bench personnel.
In this case a team would not have to play short.