A Westminister system is not really a republican form of government in the sense that the legislature elects a Prime Minister from its ranks.
How is that not republican? Indirect election of certain public officials by others has been present in many republican systems of government (including the US until 1913, lest we forget).
I suppose it actually is a republican form of government, albeit it still requires the legislature to take a lot of executive power (unless Queen Elizabeth gets off her royal fanny and starts making policy decisions
) and implants it in itself. I wouldn’t call the pre-1913 Senate a fully republican institution either, though that wasn’t what the Senate was originally intended to be in the early days of our nation. It was to serve almost like a council of elders at a congregationist church, with a sort of veto power over both the House (there is a reason why the budget is required to start in the House) and the President (through the impeachment process).
State governments have the same basic design (I believe we have a few states with unicameral legislatures). It prevents state governors from usurping too much power and becoming virtual dictators of their own states. While I strongly believe in states’ rights, I also believe that it is dangerous for too much power to be invested in individual state executives.