Your problem is an incomplete definition of a republic.
From Merrian Webster
efinition of REPUBLIC
1
a (1) : a government having a chief of state who is not a monarch and who in modern times is usually a president
(2) : a political unit (as a nation) having such a form of government b (1) : a government in which supreme power resides in a body of citizens entitled to vote and is exercised by elected officers and representatives responsible to them and governing according to law
(2) : a political unit (as a nation) having such a form of government c : a usually specified republican government of a political unit
One of the things that keeps us from being a democracy is that the US Constitution is the supreme law of the land and is rigid. The term "rigid" is used because the provisions are in a written document which cannot be legally changed with the as easily as or using the same method as ordinary laws.
The British Constitution, which is unwritten, can, on the other hand, be changed at will by act of Parliament.
No, take my country for instance, we are not a republic, for the simple fact our head of state is not elected, rather it is the Queen of Australia, and the office is a hereditary one. A democracy is a country that elects it's leaders (most democratic countries do this via representative democracy), a republic is a country with a non-monarchical head of state.
By
your definition your country would not be a democracy either. Neither your Queen nor you Prime Minister are elected directly by the people.
The Queen's position is obviously heredity.
As I understand your system, the PM is chosen by the party (or coalition of parties) with the majority in Parliament. Isn't it correct that on your ballot, your vote is for the Member of Parliament for your district?
Again from Merriam Webster: Definition of DEMOCRACY
a : government by the people; especially : rule of the majority b : a government in which the supreme power is vested in the people and exercised by them directly or indirectly through a system of representation usually involving periodically held free elections
Although the differences between a republic and a democracy may seem subtle, they are important.
The biggest of course is that in our Republic, an outcry of the people cannot bring about a Vote of No-Confidence and new elections. In your Democracy it can.
In the US, the process to amend our Constitution is tedious and difficult. This was deliberate. This is so that it cannot be changed hastily to appease an angry or discontent majority. While the process continues is debated, & possibly started, emotions can cool & saner heads will determine if it is truly necessary or just the product of an transient situation. As a result, far more amendments have been proposed than have been enacted or even sent to the states.
For the same reason the Democratic party was labelled the Democratic party, for the connotations of mob rule that go along with the word, but it is not the sole definition.
This sounds as if you believe all Presidents before Wilson and our founders were either moral cowards afraid of the connotations of the word democracy, or well meaning but ignorant of the political truth you possess.
Sorry, but that is just not the case. There was much bitter debate in the Constitutional Convention as to whether we should be a democracy or not.The democracy advocates lost the debate. The "Federalists" won the debate and our Federal Republic was formed and instituted through our Constitution.
The Democratic Party (who originally called themselves the Republican Party BTW) was founded by some of those same democracy advocates along with Thomas Jefferson, who while not at the convention (he was representing our nation in Paris at the time) vigorously supported that form of government.
The Federalist Party was formed by those who supported the current form of our Republic. They later dissolved over internal disputes & were replaced by the Whigs who were replaced by the modern Republican Party.
Wilson was just the first of many "progressive" presidents (of both major parties) who thought we should evolve toward a more democratic form of government.
The misrepresentation was repeated often enough that is has almost become a tradition, even with many who sincerely support our republican form of government.
That does not change the fact it is not true.