It's quite different. The House of Lords was much more powerful well into the 19th century. The monarch was hardly a powerless figure in those times. The Bill of Rights 1689 probably shifted the power more towards Parliament than before but the monarch was still very powerful. The UK system continues to evolve with notable precedents being set very recently like requiring a consultation of Parliament before embarking on military action and the limitation of prorogation powers.

The setup isn't the problem. The refusal to evolve is the problem.

I'd argue that it wasn't really the system in place. The system in place was one of states governing themselves. Before independence the states didn't really deal much with each other.