Automated Direct messages offend and annoy the heck out of me because this is what the senders are communicating to me:
1. You are too lazy to send a personalized message to each person.
When it's automatic, you don't even have to do any work at all! If you're too busy or get too many Followers to send individual/personalized messages, don't bother sending Direct Messages at all.
2. You have no concept of building relationships.
One of the basic principles of relationships is that people want to feel that someone cares about them. Now, who feels that way after getting an automated DM?
3. You have no concept of incentive.
You don't want to earn my traffic or Facebook Page Like. If you don't show any interest in what I do or don't even let me know why I should care about what you do, why should I do something for you?
4. I'm just a number to you.
I don't want to be part of a machine, man!
5. You think I'm an idiot.
You must think I'll feel special because you sent me a private message instead of a public one. Why else would you think that I'd click on your link in your automated message? But I know you're not sending that message just to me. Not only can I tell when the message is generic and/or automated; I know the other reason why you choose to send Direct Messages: those can be automated and sent to individual accounts; you can only @mention manually and doing so would clutter your Profile.
But not all people/companies put links in their Direct Messages because they're clever enough to know how transparent and rude that is. So they make personal-sounding generic messages to try to engage their Followers. This classic, hilarious example of an automated DM is taken from a post that ironically and paradoxically advises Twitter automatic DM etiquette:
“Looking forward to your tweets and connecting further! Enjoy your day.”
Really? You can't bother to send me a message manually or learn my name, but you think you can convince me that you're interested in my Tweets and "connecting" with me? Yeah, you read my Tweets, all right, if I @mention you or use your hashtag to talk about how great you are.
6. You never got the news that people hate spam.
When people are so used to getting it, they probably don't even look at the whole message.
7. Either the above points are true, or you just don't have a strategy.
If you haven't actually considered any of the previous points, you probably just impulsively made an automatic Direct Message, not thinking about how it would make your Followers feel, or thinking about whether the DM would actually bring you traffic.
So please, don't send generic/automated Direct Messages because it will have the opposite effect you're looking for: even when your site interests me, I'm too ticked off with you to give you the benefit of visiting it. And I know I'm not the only one.