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.
When I reported a person in my neighborhood for screaming about hurting people and herself for hours with open windows, I called the police (didn't find any email or something) and they wanted to know who I am, where I am and why I am there.
At some point I felt like they're not trusting me.
Why do they trust an email?
They can gather more information from you. They can't gather more information from an email. It forces a decision based on exactly what the message said and no more - and they'd presumably want more information in order to ignore it. For a closer comparison you'd want to call from a payphone, make your report, then hang up.
I have no idea if this would actually work similarly but it is at least what people do in movies when they want to provoke a police response! And it does make sense.
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.