You are mistaking what a countries calls itself (republic of china) with what style of goverment it has.
Ah, this is so off-topic. I didn't really mean to start a controversy with my rant. I just detest the abuse of the English language (or any language for that matter.)
Republic is not a style of government. It is a form of government which exists in many styles. One of those styles is democratic.
What is needed for a republican form of government is that no position of power be inherited by birth, but all are subject to election.
However, people can and do elect dictators. Hitler, Stalin, Mao Ze-dong, Fidel Castro, Kim Il-Sung were all elected, some repeatedly. Typically, even when a person first seizes power by force (as Stalin, Mao and Castro did) they still legitimate their rule through elections. So the republican form of government is still respected and appealed to. The dictator claims to rule by the will of and on behalf of the people.
A one-party state may also be republican in form. Although one cannot be a candidate unless one belongs to the only legal party, membership in the party is not a birthright, and the candidates must still be elected by the people.
In American history, the right to elect and to be elected was often restricted. In particular women did not always have a franchise. But the US was still a republic nevertheless. So is Switzerland, where women are still excluded from political power.
In the past, the right to vote and to stand for office was denied to many on the basis of ethnicity (native peoples of America) skin colour (apartheid South Africa) or class (only property-owners allowed to vote--19th century Europe).
These are all restrictions on democracy, but they don't violate the definition of republic. When the right to vote and the right to be elected is opened up to a wider and wider segment of society, it becomes more democratic in style. It does not become more republican in form.
That is why it is silly to speak of republic as if no republic were democratic. Many republics, including the US, are democratic. We may wish they were more so, but they are not going to be more or less republican on the basis that they are more or less democratic.
Nor are non-republics like Canada going to be less democratic because the head of state inherited her position.
In short, "republic" and "democratic" are two different terms which are not mutually exclusive. They should not be interchanged with each other. Nor should it be thought that having one means you must have or cannot have the other.
You can have a democracy which is not a republic. You can have a republic which is not democratic and you can also have a republic which is a democracy. Both conflating the terms as if they meant the same thing, or forcing them apart as if you cannot have both is an abuse of political terminology. "Democratic republic" is neither a tautology nor an oxymoron.