> To which I normally say the US isn’t a democracy, but a Constitutional Republic
It is a silly semantic debate. If you define "democracy" narrowly to mean only direct democracy, then no the US isn't a democracy. If you define "democracy" broadly, to include representative democracies with constitutionally entrenched protection of minority rights, then yes the US is a democracy, as are quite a few others.
Whether a country is a republic or a constitutional monarchy is rather orthogonal. You can have a representative democracy with constitutionally entrenched protections of minority rights, and with that you can either have a mostly ceremonial constitutional monarch, or a President. The later is a republic, the former is not.
I think the UK does fall short somewhat, in lacking a codified constitution, there is really no such thing as entrenched rights – other countries without an entrenched constitution include Israel and New Zealand. Then you also have countries like Australia, which does have an entrenched written constitution, but the rights protected by the Australian constitution are rather limited – protection against religious discrimination and the implied right of political free speech. But "democracy" vs "constitutional republic" isn't a good way of capturing those deficiencies.