_update_values also has the on_init=True flag in the @depends decorator. Do you know why that is not sufficient?

Yes, the on_init=True will cause the method to be called when the __init__ of the param.Parametrized base class is called, namely here. Since we now set the parameters after the intialization, the method is not called again.

Alternatively, a more declarative way to handle this would be:

@depends(
    "_trigger",
    "from_",
    "to",
    "start",
    "end",
    "start_derivative",
    "end_derivative",
    watch=True,
    on_init=True,
)

At the expense of more triggers. We then should also account for the intermediate triggers (for example, CubicSpline will throw an error if start > end, which could occur if start is updated before end).