Why do sports cars use active exhaust valves instead of one permanently loud exhaust pipe?

Answer: They switch exhaust flow between a quiet path and a loud path when the driver chooses.

They switch exhaust flow between a quiet path and a loud path when the driver chooses. ✓ — Right. Active valves let the same exhaust system behave differently in different modes. Closed or partly closed valves can send more gas through quieter, more restrictive paths; open valves let more unmuffled flow through for a louder note. That is why one car can be neighbor-friendly at start-up and dramatic on a back road. The sound is scheduled plumbing, not a fixed personality.

They keep exhaust pressure steady so the engine always makes its best pulling power. — Not quite. Active valves do not hold back pressure fixed; they intentionally change the flow path and restriction as conditions change. That can alter loudness, tone, and sometimes performance feel. A fixed-pressure pipe would be simpler, but it would not adapt between quiet and loud modes.

They send extra exhaust heat away so the engine does not overheat. — Not quite. Exhaust flow does affect temperature, but active exhaust valves are mainly a sound and flow-control tool here. If overheating were the central problem, the design answer would focus on cooling, materials, and thermal shielding. The valves instead change where exhaust gas goes and how loud it is.