How to sort View on rewritten field?
I think the quick answer is &quot;you can't sort on a rewritten value&quot; in Views; regardless if it's a date field or using Search API.
There are however; various workarounds depending on the particular issue. I have used the following now:

use Computed Field to store the rewritten value in the db and then display/sort on that
SearchAPI is capable of doing simple rewrites with its Aggregated Field; this may work for simple rewrites.

You can often combine a computed field and a rewrite in Views to take advantage of Views field formatting. Keep in mind that the sorting occurs on the original db value for the field. So, for my example above, you could do a computed field to store the timestamp of the Date Range's end date; and then add that to the View but rewrite that field with the formatted end date. This will then sort on the timestamp; but display the properly formatted value.