I did a test changing the Web TextField type to date. It looks like the date picker works with all modern browsers but Safari on Mac.
If you want to test, add this code to the shown event:
This is how it looks in Chrome:
This is how it looks in Firefox:
Also working and tested on my Android phone, iPhone using Safari, Windows10 Edge (older and newest).
You will read the date as SQLDate (YYYY-MM-DD) and you can change that too using that format. I guess we need to be careful, I don’t know if some action will change the type ‘date’ back to ‘text’.