Still having an issue. The example that you provided wasn't for a time picker it was for a time selector.
I have tried to do this by setting the selector style but I still only start with the current time.
<syncfusion:SfTimePicker AllowNull="True" ShowDropDownButton="True">
<syncfusion:SfTimePicker.SelectorStyle>
<Style TargetType="syncfusion:SfTimeSelector">
<Setter Property="SelectedTime" Value="12:00" />
<Setter Property="MinuteInterval" Value="15" />
</Style>
</syncfusion:SfTimePicker.SelectorStyle>
</syncfusion:SfTimePicker>
Attachment:
SfTimePicker_bcf56040.zip