When running your app myself, the package installs no problem, but indeed the time selector fails to appear on shinyapps.io while it works locally.
Firefox's developer console shows this error message:
The resource from “https://me.shinyapps.io/testpicker/_w_9415f45e/reactwidget-2.0.0/react-tools.umd.cjs” was blocked due to MIME type (“application/octet-stream”) mismatch (X-Content-Type-Options: nosniff).
Which appears in line with this report. I believe you have the same problem, that DateTimeRangePicker depends on reactR and the latest version breaks in some situations, specifically on shinyapps.io.
See the linked Github discussion for details.