This is the bane of every UI out there. "I hate the inbuilt controls so I'm going to write something better that doesn't look terrible!". Then they build some half-arsed thing that doesn't do accessibility right and only works properly on an up-to-date Chrome-based browser in a US locale with javascript enabled.
The built-in controls should always be used, not because they look pretty, but because they work the same across all websites. Users don't have to guess how this site's bespoke time-picker works (and how it doesn't). It won't fail in unexpected ways. It'll deal with your locale's date format so you don't accidentally specify the 5th of June when you meant the 6th of May.
Not everything has to be pretty all the time. Functional works, too.