Seems like a completely legitimate reason.

your son is a wussy

Sorry your for man-child.

Who’s hiding, the spider or your son?

OP: "He's more afraid of you than you are of him."
Son, and Spider (in unison): "Which one are you talking to?"
OP: *sighs* "Both, apparently..."

So? A phobia is a phobia. You cannot control them, and people of all ages and, genders and sizes can have them.

At least he didn’t burn the house down trying to kill it.

I can relate

Exactly how tight did you wrap the umbilical cord around his testicles?

That give a good reason to your son to start vacuuming and cleaning his room.

arachnophobia is real and like all phobias, not subject to age or reason.

For it to be that severe is not healthy. People tend to over react doesn't mean it is a phobia. Not growing up doesn't mean it's a phobia