Allselected function not removing row context for date hierarchy

Thanks for the insights, it is helping me to reason about this, even though your suggestion doesn't fix the problem.  If you scroll down in your example, you will find that the start date you are calculating is actually the start date for the row month, it is not constant.  Second, your method ignores any slicer on the page which defeats the purpose of the exercise.

Based on your explanation, I realised that the hierarchy, filthy hack that it is, is simply adding columns and all of these columns that are involved in the row context, need to have their filters stripped back to the visual level.

The point is, I am slicing (at page level) on 'Dim Date'[Date] but my visual is using the Date Hierarchy so I need to remove the row context from all of the hierarchy columns that are involved.  There are two ways to do this...

startDate3 = CALCULATE(
    FIRSTDATE('Dim Date'[Date]),
    ALLSELECTED('Dim Date'[Date].[Year]),
    ALLSELECTED('Dim Date'[Date].[Month]),
    ALLSELECTED('Dim Date'[Date].[MonthNo]),
    ALLSELECTED('Dim Date'[Date].[Day])
)

startDate2 = CALCULATE(
    FIRSTDATE('Dim Date'[Date]),
    ALLSELECTED('Dim Date')
)

As explained above, your suggestion, while very helpful, does not do this...

startDate = CALCULATE(
    FIRSTDATE('Dim Date'[Date].[Date]),
    ALLSELECTED('Dim Date'[Date].[Day])
)

And the results are sumarised below...

Now I have the problem though, that, because the hierarchy elements in my visual are not filtered by the slicer, I see all dates rather than all visual dates.  In other words, the filthy hack of a date hierarchy doesn't propagate the date filter to the hierarchy columns AND it is not possible to use 'Dim Date'[Date].[Date] on the slicer.  So, there is no way that I can see, to use the automatic hierarchy in a visual in conjunction with a slicer.

My solution was to create my own Hierarchy that includes the [Date] column and use that column in the slicer.

Then use the insights given by Phil_Seamark to re-write the measure targeting that hierarchy...

startDate4 = CALCULATE(
    FIRSTDATE('Dim Date'[Date]),
    ALLSELECTED('Dim Date'[Year]),
    ALLSELECTED('Dim Date'[Month]),
    ALLSELECTED('Dim Date'[Month Number]),
    ALLSELECTED('Dim Date'[Day Number])
)

or, if I don't need to be so specific...

endDate2 = CALCULATE(
    LASTDATE('Dim Date'[Date]),
    ALLSELECTED('Dim Date')
)

and these work.

The only thing to be careful of here is the sort order for the text, Month field.  You need to include a Month Number field and sort it by that...

After going through all this, the Line Graph visual insists on labeling every day at the lowest level so it's impossible to see the entire picture at day-level granularity.