Democracy: A system of government in which every citizen is entitled to participate equally in government.
Republic: A system of government by the citizens, as opposed to a monarch, emperor, or dictator.
Constitutional republic: A republic in which the powers and functionining of government are restricted by the constitution.
I don't necessarily think "democracy" and "republic" are mutually exclusive.
I do think "democracy" and "constitutional republic" are exclusive.
For example:
In a democracy, someone else's wish to confiscate my property counts equally with my wish to keep my property.
In a constitutional republic, the government's powers are limited by the constitution, providing protection for citizens against arbitrary governmental powers.
Of course, you could have a constitutional republic with a poorly-functioning constitution, which in effect functions more and more as a democracy. So there are certainly mixed cases.
Some people would resist the definitions I have given, because I define concepts so clearly. But that is the very point of concepts: to capture the essence, not to capture the mixed cases. And objective concepts are a necessary part of rational thought.
You're limiting the definition of democracy to 100% pure direct democracy i.e. literal mob rule. I struggle to imagine what such a government would even look like (on a large scale, at least).
I tend to think of a republic as opposed to a principality (it's right there in the name, even!). That is, the machine of government is thought to be held publicly rather than privately, in a republic. Suffrage has nothing to do with it, although democracy is a common implementation of a republic. (Other examples might be a military government, a religious oligarchy, or a corporate oligarchy). It is a rather broad term.
Constraining what the citizens are allowed to do, or how they are allowed to do it, via fair voting representation in their government (that is to say, limiting the powers of their duly-elected government by a constitution) does not to my mind diminish the fact that it is a democracy. Nor does it diminish if they elect representatives rather than participating in endless referendums.
My thinking on this seems clear and objective enough to me.
To me, what you're saying isn't different from saying "no, it's not feudalism, because there's a monarch".