That's an implementation issue, not a specification issue. The specification just suggests the user is shown a date picker.
That you’re correctly using html forms won’t quickly lead to browser improvements.. so the result is that users will hate your forms. Users/your customer might possibly even think that you’re to blame, and not $browserVendor.
“This site only uses functionality provided by the HTML spec” is not a useful goal in and of itself. Using the right tool for the job, which might be JavaScript, is always more important.