REF: Series.argsort should dispatch to EA.argsort

[jbrockmendel]

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the master branch of pandas.

Reproducible Example

N/A

Issue Description

cc @jorisvandenbossche unless I'm missing something? ive misunderstood the design/behavior of EA.argsort in the past.

Expected Behavior

NA

Installed Versions

Replace this line with the output of pd.show_versions()

[mzeitlin11]

somewhat xref #35178

[jbrockmendel]

gentle ping @jorisvandenbossche

[jbrockmendel]

gentle ping @jorisvandenbossche

[jorisvandenbossche]

Sorry for the slow reply here. Maybe the more general question is: why does Series.argsort not use nargsort? (which then will dispatch to EA.argsort)
Series.argsort seems like a more limited version reimplementing the missing value handling of nargsort?

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I think in previous discussions related to EAs and argsort, it was mostly about the relationship between nargsort, EA.argsort and EA._values_for_argsort (which can call which?), and I don't directly remember discussing Series.argsort specifically.
The original PR adding EA.argsort (#19957) mentions that it enables Series.argsort (although the initial end goal was sorting), but it didn't actually do that (it didn't change Series.argsort to use it).

[jbrockmendel]

Looking at this, Series.argsort looks really sketchy.

dti = pd.date_range("2016-01-01", periods=4)
ser = pd.Series(dti)
ser[1] = pd.NaT

>>> ser.argsort().values
array([ 0, -1,  1,  2])

>>> ser.values.argsort()
array([0, 2, 3, 1])

>>> pd.core.sorting.nargsort(ser.values)
array([0, 2, 3, 1])

What would the ser.argsort() be useful for? Passing it to something like take produces weird results (which vary depending on which take-like thing you use).

[jbrockmendel]

Once the deprecation in #54219 is enforced, we will be able to refactor in such a way as to close this.