Hi @DJ_Kim ,
Thanks for the log messages. I can see that it is not the date formats preventing the graphs from being interactive. It is the X Group's "Levels in View" setting.
The relevant log message is "Interactive HTML: Graphs with group levels in view set are not interactive."
From your image and description, I figure that you have set "Levels in View" to 2 to compare graphs of two years of data out of possibly more than two years of data collected. In JMP, you can click on the gray triangle pointing to the right in the top right corner to advance the DATE to the next year.

This has not been implemented in JMP Live yet, but you can use a local data filter on the DATE column instead of using "Levels in View" to pick two years to compare. It is not as convenient since it will require two clicks instead of one to change dates, but it should be interactive as long as "Levels in View" is set to "All". It will give the viewer the flexibility to pick more than two years or pick two years that are not consecutive.
I also see in the log messages:
"Interactive HTML: Local Data Filters with the Include mode enabled are not supported in stand-alone Interactive HTML"
This suggests to me that you are already using a local data filter. All you will need to do is add the DATE column as another column to filter on. The message is given when you export to Interactive HTML because without the power of JMP in JMP Live, stand-alone Interactive HTML files cannot exclude data, they can only hide data using the local data filter.
Here's a script I used with the Diamonds Data.jmp sample, to show how the local data filter can be used on the Clarity column, which is also used in the Group X role to display two(or more) graphs of different Clarity levels instead of using "Levels in View" to limit the groups to show only two levels at a time. It will be Interactive when published to JMP Live.
Graph Builder(
Size( 654, 486 ),
Show Control Panel( 0 ),
Variables( X( :Cut ), Y( :Price ), Group X( :Clarity ), Color( :Cut ) ),
Elements( Contour( X, Y, Legend( 16 ) ) ),
Local Data Filter(
Add Filter(
columns( :Clarity ),
Where( :Clarity == {"VS2", "VS1"} ),
Display( :Clarity, N Items( 7 ) )
)
)
);
Here's a screenshot of it on JMP Public (a public instance of JMP Live).

You can try this interactive example on JMP Public at : JMP Public example using a local data filter instead of Levels in View .
Please let me know if this works for your case. If you would like us to add support for "Levels in View" in JMP Live, please add a Wish in the JMP Wish List.
Thanks,
~John