To your last question, no DNA test is going to yield "actual" matches unless the people with whom you match have also done the test and are in the same database you are using. Even then, you have to pay attention to the probabilities – they might not be actual matches, and the probability of that being the case increases the futher back you go.
To your more general question, autosomal is not the tool to use the find a specific ggp 19 for various reasons. I have to state again, it can show that you share some DNA markers with another person indicating you *might* have a common ancestor somewhere, but it won't tell you where they are in your tree. By analyzing trees from multiple matches, you might be able to figure that out, however, you don’t know if those two people are matching you for the same reason (hence the call for getting the raw data). The problem you have is the vast majority of trees I've seen don't go back anywhere to 19ggp, and if they do, it's only on one or two lines. Even when they do go back that far, it's because the person was relatively famous or well known so there is documented evidence that scholars before you have researched. Anyone who has that same info in their tree probably got it from the same place you got yours. The vast majority of people who lived in the year 1100 left no evidence of their existence.
As far as what to use, if that person is a direct male ancestor of yours, then you can do Y-DNA which gets passed down relatively unchanged instead of autosomal which gets "diluted" each generation. If you are not and you know somebody who is, then they need to do the test. Even with that, the results will only help if others in that line have tested and are in the same database you are part of. The way to help with that is to join a surname project. Likewise, if this ancestor follows a direct female line, then you can use mtDNA. If on the other hand ggp 19 is along some indirect line like father's mother's mother's father etc., then you could use autosomal to track down that 4th cousin in Montana who might be a direct male or female descendant.
As I said in a prior message, the value of DNA testing all depends on your goals. I understand that you and others are interested in following some of your lines way way way back – that's not uncommon, especially with someone's surname. In my case, I'm more interested in following all my lines back maybe 200 or 300 years, so those 4th and 5th cousins are very useful. My logic is once you get further back than that, my relationship to any specific ancestor would have little impact on my family history, plus it's not that unique because that person would have thousands and thousands of descendants. Also, I’m just as related to ancestors who don’t carry my surname, and in fact, because of several incidents or relatively recent intermarriage (2nd cousins) I'm actually closer related to ancestors on my maternal grandmother's side than the other 3 lines.
Since you've already done the autosomal test, I'm sure you are matching on people closer to you (you mentioned a 3rd cousin for example). That information can be used to help with more recent generations. For example, I would be surprised and impressed if you know for sure all 64 of your gggg-grandparents.