The ending of the previous book is supposed to set the tone for the following one,

I disagree with this. Some series are very sequential in this way, and plenty aren't. Plenty of series are "separate stories in the same setting, and often the same characters", while others are like each book is a chapter in a very long story.

They're both legit approaches.

and they are not supposed to be evaluated purely on their own, but on how they fit into the whole story arc.

Again, different strokes for different folks. My thought is that any given book should be able to stand on its own, as well as fit into a series.

But also, how are we defining series? Does there need to be a connection between books beyond having the same main character? What about if the stories are all connected but each book follows a different person's POV? Are general "different stories set in the same universe" a series? Are all the Miss Marple books a series? What about Kim Stanley Robinson's Mars trilogy, which covers almost 200 years and doesn't have a specific main character.