BUG: set_index with pyarrow timestamp type does not produce DatetimeIndex

[WillAyd]

Pandas version checks

• 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 main branch of pandas.

Reproducible Example

import io

import pandas as pd

buf = io.StringIO("date,value\n2024-01-01 00:00:00,1\n2024-02-01 00:00:00,2")
df = pd.read_csv(buf, parse_dates=["date"])
df.set_index("date").loc["2024-01"] # works


buf = io.StringIO("date,value\n2024-01-01 00:00:00,1\n2024-02-01 00:00:00,2")
df = pd.read_csv(buf, parse_dates=["date"], dtype_backend="pyarrow", engine="pyarrow")
df.set_index("date").loc["2024-01"]  # KeyError

### Issue Description

The pyarrow timestamp type gets put into a generic `Index` when assigned via set_index, so the datetime overloads do not work correctly

### Expected Behavior

The pyarrow timestamp type should be wrapped by a DatetimeIndex

### Installed Versions

3.0.0.dev0+1696.gfae3e8034f'

[WillAyd]

I think this is another one to keep track of for PDEP-13 #58455

[AbhishekChaudharii]

take