Radio Antenna Fundamentals Part 1 -- 1947 USAF Film

Lots of good info in this 1947 film.  But... At the end they say that an impedance mismatch between the line and the antenna will result in the loss of radiated power.  Is that really true?  This seems to lead to the obsession that many hams have with an SWR of 1:1.  If some power is reflected due to the mismatch, where does that power go?  And what if you put a transmatch at the junction of the transmitter and, say, an open wire feedline.  You could match the antenna system perfectly to the 50 ohms that your transmitter is looking for.  But the mismatch between antenna and feedline would remain, right?