I've done some testing and I think there are a couple of issues at play here.
Based on your third image it appears the roundabout is a single line that starts and ends at the same point (self connecting/crossing). Each leg of the roundabout should be split to its own segment to provide a distinct network edge. If you create the roundabout as a single line where the first and last vertex/node are at the same point, one of two things will happen depending on how the network was set up (this can be changed by editing the network dataset properties).
- The connectivity policy for the street layer is set to End only: If
your first/last roundabout point happens to also be an intersection
with one of the legs, you can get from that leg onto the roundabout,
but not off because a junction can only occur at a line endpoint
(aka node, or first/last vertex).
- The connectivity policy for the street layer is set to Any Vertex:
Junctions can now occur at any vertex, and routes can be found
around the line. However, no network edge will be created between
the second-to-last vertex and the first/last node depending on how other lines intersect the roundabout.
This network connectivity policy is easiest demonstrated by creating a T intersection. Create a line with three vertices, then a perpendicular line with two - one of which snaps to the middle vertex of the first line. Now create your start point on one leg of the T and your stop on another. In case 1 above, the route will fail with that error message. In case 2 the route will succeed. I can't really explain why your one pre-existing roundabout is functioning, but I cannot tell how it was drawn (ie where the nodes are or if there is a network edge all the way around).
You may also need to modify (or model in the first place) one-way and turn restrictions to ensure correct flow onto, around, and off the roundabout. Your first image shows an invalid route based on the background map arrows. Note that the network must be rebuilt any time the underlying layer is changed.
It may help to turn on the network junctions and edge layers, then use the Network Identify tool to examine the pieces. When you click on a junction or edge, it will tell you which other edges or junction are adjacent to the selected (thus you can see what is and isn't connected).