The problem with customer service being 'separate' is that it quickly loses value. In that sense, OP is correct but one needs to take it further.
Once you support a customer, you are supposed to help her for more than just 2 days! You need to take it to conclusion regardless of who you are, even if it means that it takes days or weeks. The best part of that - you build a relationship with that person and if its a B2B thing then you could even add them on LinkedIn, right? ;-) Heck, you could even do some Marketing/Sales for your company and ask them for a testimonial about how good the customer service was! There are even tools for such things - Customer Rivet (<a href="http://www.customer-rivet.com" rel="nofollow">http://www.customer-rivet.com</a>), or Testimonial Monkey (<a href="http://www.testimonialmonkey.com" rel="nofollow">http://www.testimonialmonkey.com</a>) etc..
End of the day, what I learnt (the hard way) is that you need to support customers not just because its company policy but because it helps you out personally.