Not only that, but having a democratic constitution would constitute socialism.

A democratic constitution was also one of the goals of the early communists - but again, a country is not communist until it has fully implemented communism.

Also, keep in mind that while Communists wanted to use socialism as a way to reach communism, that does not mean that for many people socialism is not an end in itself (there are plenty of socialists - as opposed to communists). It also does not mean that all forms of socialism are consistent with a path to Communism - they are not.

The early communists identified democratic socialism as the movement that they could work with to reach communism.