It is a democracy. A Constitutional Republic is just one way of implementing a democracy, similar to how a constitutional monarchy is just one way of implementing a monarchy. Mob rule everyone votes on everything and even a majority by one decides everything is another (poor) way of implementing democracy.
Sorry... your logic is flawed at least semantically. If a "constitutional monarchy is just one way of implementing a monarchy," then presumably a "Constitutional Republic is just one way of implementing a" republic.
The only nexus between republic and democracy in this case would be the Constitutional Republican Guarantee clause. Since that has essentially never been adjudicated we can't know for certain whether - at least in Constitutional terms - a Republic does convey democratic rights.
