Hey Brandon,
Come sell for me! Just kidding. I do like your straight forward approach though. But, with link building I would suggest first changing your point of view just a bit.
First, there are many ways you can assist a webmaster and acquire a link. What you did is called a reciprocal link. "I will give you one Brandon, but I want one in return." So, reciprocity. Those links are frowned upon by Google and are said to be of little value. (Yes, I am saying that is what "they" say.) But, if you put a link in your author info or in your content and someone picks it up, and later down the road for whatever reason you happen to link to that same site from yours as it answers some point you are making, it IMO is still a natural link both ways. (Larry nor Sergey will tell me though.)
Then you say this: **What I discovered was that they displayed random links from the "Useful Links" directory in the sidebar and in the footer. So all of a sudden we have a ton of links from one website. This site has hundreds maybe even thousands of pages. **
So, the value of the sidebar and footer links is less by far than in the content portion and that is likely helping you. But, personally, without looking at a business reason for not doing it, I would disavow all those links as preventive medicine. (Those who want to watch the world burn slow down a second and I shall 'splain my point.) My reasoning is twofold, first when I think of a site having natural links and I ignore whatever definition of natural that is in vogue, I look at a site/page that for whatever reason gains some measure of "popular" traffic. If, people are there for some reason (traffic) and they are being driven for any honest reason (not sent because they thought they won $ from an email), some percentage of them will "share" that find or that info, etc. (I sent the video of the guy pushing his girlfriend off the cliff to several people when I first saw it on CNN.) In SEO, a lot of what we do is centered around mathematics and probability, other is around behavior, other is creative, etc. But the probability portion provides that at some level, as something becomes more and more popular, more and more people will want to show it to others (link), NATURALLY.
At the same time, lets take a site that has nothing really drawing traffic to it. It is there and gets X visits a day and occasionally more, but not a lot. If you look at its link growth, it should mirror the traffic to some degree. But, if all of a sudden it appears to start gathering link steam, without gathering traffic, I have to believe that in itself could provide a strong signal as to "natural" or not.
I said that to provide a predicate for this: If you build quality content that people want to read, you will gain links. As more people come to the site, find it as a resource, etc. more link to it. Over time more links bring more traffic and more traffic brings more links. What could be more natural.
I can tell you from experience it works and that we still have certain link building initiatives for sites. Occasionally, I will allow someone like you a link just because I want to. I do not ask for a reciprocal link. I can tell you that if you want to write something that fits for one of our sites and the content is good, I will give you the link without blinking. Just make sure it is not you are writing an article for a veterinarian site and linking back to a gambling as that affects my traffic.
I hope this helps and, hell, if they read this I will thumb them up.
Best Brandon, Good luck,
Robert