There is so much here to discuss... But lets first discuss the term "Catch and Carry". NFHS rules do not allow catch and carry. Catch and Carry is an OBR rule. The NFHS rule is commonly called "Straddle". If a player opts to catch and go into dead ball territory the dead ball award is in play...
Most dead ball areas in HS baseball are for safety reasons...made to keep a player from running into fences, or drainage or out of the field of play. Once an area is identified as a Dead ball area, the applicable award for entering is in effect. As an aside, Umpires dont "give" bases, the rules "award" bases. Small point possibly, but it is important.
Lets go to the rule:
NFHS rule 5 Section 1 Article 1 (i)
Ball becomes dead immediately when:
I: a fielder, after catching a fair or foul ball, leaves the field of play by stepping with both feet over any boundary or barrier such as a fence rope, chalk line, or a repgame determined boundary line.
So your player had the option of trying to make the catch with one foot in dead ball area to record the out and not violate dead ball space or let the ball drop foul.
As to changing the rule, the NFHS is constantly changing HS rules. But in this case, there is no reason why dead ball areas would be considered differently depending on where they are located, since the safety issue is the same no matter where the area. As a matter of fact, most of the serious dead ball areas I know of are in and around the infield or dugout areas...
Lastly the rule did not penalize the defensive team for entering dead ball area, they chose to enter the area to record an out. Possibly some coaching from the coaching staff seated in the dead ball areas might have been a better choice....the coach could have decided if the losing the run to get the out was a better baseball decision....