There are compressive forces. If mass inside the ring is not balanced, it can drag the ring into an ellipsoid. The inner sides of the ellipse are compressed.
A rigid ring can resist some of this inherently, but a rigid spoke to the hub cleanly takes up all the inward forces.
If your ring is not rigid, any perturbations can cause oscillations that throw the whole thing out of balance. Like a gas leak in one compartment adding thrust at a weird angle. Soon the whole ring will be oscillating along its plane, which is obviously bad. You can actively correct with thrusters on each segment, but that's a lot of extra complexity.
Basically it's all about stability. A big rigid object is much harder to shake apart. A metal circle will stay a circle in a lot more circumstances than a circle of rope will. Doubly so when rotating in zero gravity.
Flexible tethers are mainly good for small scale. Swinging a crew capsule about a big mass (Project Hail Mary, Stardancer) is indeed cheap and easy. With the complication that you must completely spin down to maneuver or dock.
I don't think your reasoning here is correct. Take a bicycle wheel for example. The spokes on a bicycle wheel do not take any compressive forces due to the way they are attached to the rim, and yet bicycle wheels can take surprising compressive loads without going oblong.
A rigid ring can resist some of this inherently, but a rigid spoke to the hub cleanly takes up all the inward forces.
If your ring is not rigid, any perturbations can cause oscillations that throw the whole thing out of balance. Like a gas leak in one compartment adding thrust at a weird angle. Soon the whole ring will be oscillating along its plane, which is obviously bad. You can actively correct with thrusters on each segment, but that's a lot of extra complexity.
Basically it's all about stability. A big rigid object is much harder to shake apart. A metal circle will stay a circle in a lot more circumstances than a circle of rope will. Doubly so when rotating in zero gravity.
Flexible tethers are mainly good for small scale. Swinging a crew capsule about a big mass (Project Hail Mary, Stardancer) is indeed cheap and easy. With the complication that you must completely spin down to maneuver or dock.