There are a variety of date/time pickers out there, but honestly, it depends on your requirements. Tempus Dominus works fine with a little configuration, but it has some issues, I agree.
If you need to support older browsers, then you'll need to use a package that supports modern browsers, and then user the user agent to determine whether to just show a plain datetime input or the package input etc.
Honestly, I stick with modern date/time pickers, I have no time for supporting older browsers anymore, so almost any vue / bootstrap 4 datepicker you come across will be fine, if that is your approach.