The AliExpress transmitter does not have an amplifier. Its sales sheet says it is powered by 5V. One Li-PO battery cell produces 3V (dead) to 4.2V(fully charged) which is not enough. The extremely cheap voltage booster (they say it is an elevator) is Chinese with very confusing badly translated instructions to change the 12V output it comes with to the 5V you need by soldering "replacements but not alternations". Maybe you must remove two solder drops.
EDIT: AliExpress has a transmitter for charging a phone and has a wireless charger for charging a phone. They are both powered from 5VDC.