It's very unlikely that it's true cut 'n paste in the responses. Instead Blizzard appears to be using a CRM system with very little oversight.
For a number of years at my last place of employment, I was the manager of the team that handled customer service and support emails. When we started out, it was just two of us who basically just went through assorted email accounts via Outlook and copied/pasted appropriate responses. It didn't take long before it became more than obvious we needed a proper tool for that kind of work and the company invested in an automated CRM system.
The way those systems work is that one or more administrators spend a good chunk of time creating various queues and rules to channel incoming tickets and email messages to the appropriate person(s) and suggest a response. So, basically, a customer submits an online form or sends an email to some support address. The CRM system runs it through the rules and filters and determines who should handle it and puts it into that person/team's incoming mail queue. If certain conditions are met, the system will suggest a pre-written response. When the support person logs in and starts processing the emails in their queue, they'll see the incoming message along with the suggested response.
Now what is *supposed* to happen is the support person reads the customer's message and does whatever needs to be done, then reviews the suggested response to make sure it's appropriate. If it is, he/she customizes it a bit by doing things like adding the customer's name and other information specific to the request, then sends the response out. If the suggested response isn't appropriate, the user rejects it and either selects another pre-made one that is or creates one 'manually.'
At some regular interval, a supervisor or manager or whatever will log into the system and review random responses that the support people made for quality assurance purposes. Also, at regular intervals, productivity reports are run to see how many customer requests/tickets each support person is answering and how quickly and such.
That's what's supposed to happen. In Blizzard's case, it seems obvious that it doesn't happen, though.
In Blizzard's case, it seems the support people are more quota driven than quality driven. Just like in a call center, the employees are expected to handle a certain number of tickets per hour. Now, there's usually some leeway given with quotas since how many tickets you process depends upon the type of ticket. In the case of my old employer, tickets ranged from super fast things like "remove me from your mailing list" to far more complex ones that could take a long time to research and take care of. I suspect Blizz has set a very high quota just so that their queues don't get all backed up and doesn't take into account the difficulty of the tickets processed by the support people.
I also don't think there's any real oversight or QA going on. Basically, the high-ups are only concerned that the support team processed X number of tickets per hour and that they don't bother going and actually spot checking to make sure that the tickets are being processed correctly.
Where I worked, it took a year or two to convince the upper management that it was better to have more people spending a little extra time on each request than it was to go with bare minimum manpower and strict, high quotas. We did, though, and I do believe it paid off in the end. We tended to have a good bit of positive feedback from customers when it came to customer support.
Blizz seems to have taken the opposite approach. They just want those tickets closed, so some overworked GM who is trying to make sure to make his/her quota just sees the system suggested a response about phishing and hits Okay. That's how you get the annoying inappropriate response emails. It can happen under a well managed system too, but it's a lot more rare.