Of course an email is easy to fake, but you really have to put things in perspective: We, as internet regulars, hear of swatting quite often, but for your average police officer, this is probably an once-in-a-career type of event and therefore the threshold for believability is quite low.

Also, as you pointed out, if it turns out to be a real situation, it will look horrible if the police knew of it beforehand. Even more, a victim in a situation that requires a swat team might have a quite limited ability to communicate, lowering the reasonable threshold even more.

Lastly, there isn't really a measured response. If you think there might be an actual dangerous situation, sending a few normal officers to check it out is reckless at best and might actually make the situation worse, as the attacker is alerted of the impeding police response and the police doesn't have the ability to respond swiftly.