I think your question speaks to something deep and interesting about XPBD.
All classical mechanics sims revolve on some level around "integrating" Newton's second law of motion: F=ma. Force equals mass times acceleration. XPBD constraints correspond directly to forces, and acceleration is the second time derivate of position. So to update positions every frame, which is what we want our physics sim to do, we need to integrate F=ma.
Usually we want to use an integrator in the family of "unconditionally stable" integrators. These integrators are guaranteed not to blow up, no matter what weird, difficult input the user gives to the simulation. In realtime physics sims we generally use one called "Implicit Euler." And Implicit Euler is the worst one! It's the least accurate and it adds huge amounts of artificial damping, but it has one really big thing going for it. It's the most resilient to "under-converged" solutions to the equations of motion.
So, like you pointed out in your question, if you resolve the constraints one after the other, constraints further down the list are likely to violate previous constraints. Most of the time you can mitigate this by running over the list of constraints multiple times, but you'd have to do that many many times for your constraints to perfectly "converge," such that they are all simultaneously satisfied.
The more accurate and energy preserving unconditionally stable integrators, like "Implicit Midpoint," require perfectly converged solutions. If you don't resolve your constraints perfectly, the "unconditionally stable" Implicit Midpoint will explode! But good ol' Implicit Euler is much less up tight. You can give it surprisingly under-converged solutions and it'll keep on trucking. The shortcomings of Implicit Euler are reduced by taking smaller timesteps between frames, so you generally get the best bang for your buck by investing compute time into more frames with a sloppy, non-converging solver, than using a better integrator that forces convergence.
At least that's what it looks like now. There's still tons of active research going on in these areas. Maybe someone will discover a faster solver that makes convergence easier, in which case we'll all switch over to move sophisticated integrators.
All classical mechanics sims revolve on some level around "integrating" Newton's second law of motion: F=ma. Force equals mass times acceleration. XPBD constraints correspond directly to forces, and acceleration is the second time derivate of position. So to update positions every frame, which is what we want our physics sim to do, we need to integrate F=ma.
Usually we want to use an integrator in the family of "unconditionally stable" integrators. These integrators are guaranteed not to blow up, no matter what weird, difficult input the user gives to the simulation. In realtime physics sims we generally use one called "Implicit Euler." And Implicit Euler is the worst one! It's the least accurate and it adds huge amounts of artificial damping, but it has one really big thing going for it. It's the most resilient to "under-converged" solutions to the equations of motion.
So, like you pointed out in your question, if you resolve the constraints one after the other, constraints further down the list are likely to violate previous constraints. Most of the time you can mitigate this by running over the list of constraints multiple times, but you'd have to do that many many times for your constraints to perfectly "converge," such that they are all simultaneously satisfied.
The more accurate and energy preserving unconditionally stable integrators, like "Implicit Midpoint," require perfectly converged solutions. If you don't resolve your constraints perfectly, the "unconditionally stable" Implicit Midpoint will explode! But good ol' Implicit Euler is much less up tight. You can give it surprisingly under-converged solutions and it'll keep on trucking. The shortcomings of Implicit Euler are reduced by taking smaller timesteps between frames, so you generally get the best bang for your buck by investing compute time into more frames with a sloppy, non-converging solver, than using a better integrator that forces convergence.
At least that's what it looks like now. There's still tons of active research going on in these areas. Maybe someone will discover a faster solver that makes convergence easier, in which case we'll all switch over to move sophisticated integrators.