As a consumer, if a company with a Twitter presence responds that way I'll ignore them. Because 9 times out of 10 when I tweet at a brand it's either because I'm calling them out for actively harmful stuff (like using plaintext passwords) or prodding them about issues that likely affect more people than just me.

If I have a specific support request about my personal account, I'll go through the usual channels. But if it's on twitter, it's probably there because the extra visibility and public accountability is intentional.