BUG: StringArray.astype() raises for datetime, period, boolean & timedelta with missing values
- [X] I have checked that this issue has not already been reported.

- [X] I have confirmed this bug exists on the latest version of pandas.

- [X] (optional) I have confirmed this bug exists on the master branch of pandas.

---

**Note**: Please read [this guide](https://matthewrocklin.com/blog/work/2018/02/28/minimal-bug-reports) detailing how to provide the necessary information for us to reproduce your bug.

#### Code Sample, a copy-pastable example

```python
import pandas as pd
s = pd.Series(["2020-10-10", None], dtype="string")
s.astype("datetime64[ns]") # fails

s = pd.Series(["1 Day", None], dtype="string")
s.astype("timedelta64[ns]") # fails

s = pd.Series(["1/1/2021", None], dtype="string")
s.astype("period[M]") # fails

s = pd.Series([True, None], dtype="string")
s.astype("boolean") # fails
```
Similar issue for `timedelta64` raised by jbrockmendel in #38509. 

#### Problem description
The behaviour looks like a bug and I'd like to help out, however the `astype` dispatching (https://github.com/pandas-dev/pandas/issues/22384) and the `_from_sequence` specifics (https://github.com/pandas-dev/pandas/issues/33254) seem to be debated issues.

If you'd like I can write a PR that fixes this by:
1. implementing `_from_sequence_of_strings` in the ExtensionArrays where it hasn't been implemented: Categorical, DatetimeArray, TimedeltaArray.
2. in `StringArray.astype()` convert to ExtensionArrayDTypes using `_from_sequence_of_strings` instead of `_from_sequence`.



#### Output of ``pd.show_versions()``

<details>


INSTALLED VERSIONS
------------------
commit           : f2c8480af2f25efdbd803218b9d87980f416563e
python           : 3.8.6.final.0
python-bits      : 64
OS               : Linux
OS-release       : 5.8.0-45-generic
Version          : #51-Ubuntu SMP Fri Feb 19 13:24:51 UTC 2021
machine          : x86_64
processor        : x86_64
byteorder        : little
LC_ALL           : None
LANG             : en_US.UTF-8
LOCALE           : en_US.UTF-8

pandas           : 1.2.3
numpy            : 1.19.2
pytz             : 2020.5
dateutil         : 2.8.1
pip              : 21.0.1
setuptools       : 52.0.0
Cython           : None
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : 2.8.5 (dt dec pq3 ext lo64)
jinja2           : None
IPython          : 7.19.0
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : 0.8.7
fastparquet      : None
gcsfs            : None
matplotlib       : 3.3.4
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : 3.0.0
pyxlsb           : None
s3fs             : None
scipy            : 1.6.0
sqlalchemy       : None
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None
None


</details>
