When I asked the host again, I got a different support tech who said:
All our servers are running like this due to server abuse in the past
runing in cgi mode means we can watch everything that is going on
with the server in su-exec mode
sorry it is not possible to change this
Thank you for that advice about the $_SERVER superglobal array variable.
I also found a potential workaround (untested) here in the first comment that follows the text. Not being proficient in PHP it's time to eat some books methinks.