Is there a systematic theology resource based solely on the words of Jesus?

I don't think there's a satisfactory answer to the original question, but I thought an answer would be better suited than a comment.

There can't be a total solution to your question, because Jesus was constantly making reference to the OT Scriptures. So, you cannot have an only - word - in - red approach, as it would be self-contradicting. And the life of Jesus was in the context of Jewish culture, the temple, etc. He was Jewish.

But, while the goal would seem to be to locate the authentic from Jesus' teaching directly, there are a few this to consider in this approach. many have claimed to do this, but here's some things to watch for.

• At a basic reading, Christ's life was about the miraculous. Regardless of doctrinal position, you need to address this somehow. If you accept it for today, it should be reflected in the life. If you reject it as for today, you lose out on a large portion of the Gospel instructions and example. So, how you deal with miracles, demons, and this are hugely important.
• Second, many attempt to create a "Social Gospel", or other works based mentalities, looking only at outward works. As this contradicts a salvation by grace through faith, it is contrary to orthodox (normal) christianity. If the teaching methodology is more about outward behavior than an inward heart, it is to be discarded (in orthodoxy).
• Deal with faith. Faith is not a set of beliefs. Jesus defined faith in Mark 11:23-24, and if your definition of faith doesn't match his, you aren't doing your original premise. James says faith without works is dead.
• What about the hard passages? Many people love Jesus' words about helping the poor. What about hating your own mother? And your own life, too? Many want only to deal with the "gentle" passages.
• Deal with the extremes. Jesus said if you deny him before men, He will deny you. He said not to fear him who can kill the body only. He said calling your brother a fool can put you in hell. He said simply calling Him LORD is not enough to be saved.

All these points tie together, and work together towards the truth. Like different facets.

But, what have I done? I have now talked on and on about the "Words in Red", using something other than the words themselves. If this were divinely inspired, someone could put them in a book, bind them, and consider it the authority on the Words in Red and how to live them. See where I'm going?

The epistles unpack the teachings of Jesus. They can be considered divinely inspired sermons, and they give practical application to the principles involved in the Gospels. So, in excluding them, what have you done? You take the most inspired and true exposition and teaching trying to explain all those lessons, and you throw them away, and what you end up doing is just listening to some other preacher instead.

So, a possibility would be to embrace the whole NT as inspired, and, if you choose, steer away from interpretations of the epistles that minimize the Gospels. Then, examine how living as it is described in the epistles is walking out the instructions of Christ, or vice versa.

The benefit of such an approach will be you will need to know what both of them say, and mean, and this is at the heart of biblical discipleship. And, that's the real point. History is full of examples of people lifting one or two favorite points out of Jesus' words, but, by their fruits, they missed what He was saying by miles. But, the doctrine of the church at large is that Paul did pretty good (evidenced by the fact that we read his stuff!).

I'm pretty sure that there is no source that looks at only Jesus words. However there are certainly theological approaches that place a much greater emphasis on Jesus words. For a start you might want to look at Red Letter Christians, a group that attempts to redress what they see as an imbalance in North American Evangelicalism, which they believe has seriously de-emphasised Jesus' words in favour of Paul's writings and selected Old Testament teachings. However the members of that group do pay attention to other parts of the Bible, and do treat them as inspired and authoritative.