Using the date input field should always adhere to the UX of the browser OS.
If I'm on Android and I spend my whole life learning the UX of Android's date picker, wouldn't you want the web's date picker to follow the same UX?
This is less a problem the website should solve and more of a problem that end-users need to learn.
If my grandma doesn't know how to use a feature on her phone you bet she is contacting a family member first. Maybe we should encourage our family to use tools that they are proficient in...
I'm an iOS user majority of the time and it is very intuitive to tap the text with the > symbol. Like others state, Android's UX can be just as confusing.
I'd say you're following UX best practices and are supporting the native accessibility features of the phone so :shrug:... close and move on