Why has my tweeter speaker burned up?
Your linked description shows that the tweeter is intended to be used with a woofer for music reproduction. The description doesn't cover use of the tweeter as a specialist ultrasonic transducer on its own. The power rating indicates that the tweeter is intended to be used with a 350 W average/700 W peak audio amplifier driving both tweeter and woofer. For normal music or speech the average level is 10 dB below the peak level, bringing the peak level down to 70 W, much of which is feeding the woofer. In order to ensure the bulk of the power actually does go to the woofer the tweeter impedance is, as you found, much higher than 8 Ω. The  4-8 Ω rating in the specification indicates that the tweeter is intended to be used with a 4-8 Ω woofer, not that the tweeter itself has a 4-8 Ω impedance.
The Pyle description is written to be easy to understand for someone with modest technical knowledge putting together a car audio or other speaker system. The description tells them that if they have a 350 W amplifier and a 4-8 Ω woofer then this tweeter is suitable to use with them. It doesn't mean that you can put 350 W into just the tweeter.