You have to enable "debug mode",
Then open browser console on "broken page", and check for errors there

I don't see any JS error. I think some script that manage those calendars (make them disappear and appear only when clicking on the icon) is missing.

the calendar script does not have extra dependency, if I right remember

There is at least one CSS issue: when clicking on the Close button on the calendar, it adds the class "hidden" but it is not something bootstrap understands. https://getbootstrap.com/docs/5.0/helpers/visually-hidden/ or maybe "d-none" for better accessibility.
I guess it is defined somewhere in Cassiopeia but not calendar specific css.

Another issue is the button Remove item not properly styled

Another CSS fix:
calendar.css defines
.calendar-container {}
with background white but does not define the color. Containers that define the background color, should also define the text color because my body text is white (just defining bootstrap colors sass variables to be a dark theme) so it appears with white text on white background.

because my body text is white

You can and you should provide your own CSS override in your template for the calendar (and many other core parts). The default CSS was written with the assumption that Protostar would be the template used.

Please test #34871