You have the code there to support it, it's just not been put in the correct order. You need to build your date and update the content
within the event listener code. For example:
time_zones.addEventListener('change', () => {
const timezone = time_zones.options[time_zones.selectedIndex].value;
const date = new Date().toLocaleString("en-US", {
timeZone: timezone,
timeStyle: 'medium',
hourCycle: 'h12'
});
});