What weakness would this expose? It doesn't seem obvious that there would be any apparent way to attack.
This may seem strange but I believe doing that actually makes the system stronger. Using the same path always prevents an attack called a predecessor attack. It also makes end to end correlation attacks more difficult as it only gives the adversary one shot at being selected by the client to be in their path.
This "it's actually more secure" line of thought is the reason guard nodes exist. If you don't know guard nodes are the first hop in your circuit. When a client enters the network it selects a set of relays to be its guard nodes. A client's guard nodes stay the same for as long as possible.
This makes your entire traffic highly fingerprintable as compared to a standard random path. If your connections always used A, B, and C nodes, it is statistically unlikely that many other people are consistently using that same path, therefore it's very easy to correlate your traffic to your originating IP.
An adversary exploiting this would need to have purview of each step in the connection to find out that you're using this static path. But once its been found, they would be able to make a simple 1:1 correlation of the traffic by watching when you access the Tor network, and then watch your exit node that never changes. (As opposed to you taking a variety of paths and choosing a variety of exit nodes.)
While the idea of guard nodes (static entry nodes) offers more protection, making the entire route static would reduce your anonomity.
In terms of traffic correlation attacks you've in some ways reduced Tor down to a single hop proxy.
If everybody uses the same path, it will be like JAP. However, if everybody uses different paths, timing attack will be easy. Performance is also a problem.