Your cast query is correct but I believe the source timestamp formatting cannot match the cast's accepted format. The closest I can get is using parse_date function. But I have extracted the the date only using string manipulation(ragexp_contains) and extract the year to current date to full fill the formatting of the parse date paramter.
SELECT
PARSE_DATE('%b %e %Y', CONCAT(REGEXP_EXTRACT('June 13, 7:23pm EDT', r'^(.*?),'), ' ', EXTRACT(YEAR
FROM
CURRENT_DATE()))) AS date
It is possible that there is a more optimal approach to this but here is the ouput of the query:
