to a point you are right, but clearly there is some boundary, right? imagine a solid object with a mass just below the Chandrasekhar limit and a temperature near absolute zero. Don't tell me hurtling that into the Sun will make it burn hotter. And even if it would (or a smaller object) in the long run, there would at least be some period of time before the incoming matter reached fusionable temperatures, reducing power output in the short term.