I personally found that keyword density is somewhat rather irrelevant. I never even check it or ask copywriters to maintain a certain % of keywords in the text. Instead, I ask them to ensure the best readability possible.
Contrary to popular belief that you shouldn't be doing multiple articles with the same keyword to avoid competing against youself in SERPs, I found that doing multiple articles with the same keyword enables synergetic effect. It might be that multiple articles with the same keyword signal Google that this keyword is one of the main website topics.
I think most SEO specialists overestimate the importance of parameters that can be manipulated, e.g. kw density. I belive one of the most important factors (if not the top factor) is how many people return from your website back to Google. The less do so, the higher Google will put you in SERPs.
To give you another perspective, try the following imagination game. Let's say, you need to collect the best possible news on the topic of neural networks. You have two choices:
1. Receive feeds of 5 best websites dedicated to neural networks to your inbox and read them daily
2. Receive daily reports about what's happening in the field from 5 best scientists in the world working on neural networks
What do you think would be more effective approach information-filtering wise? I bet you chose the second one. And I bet Google thinks identically. This is why algorithms based on human behavior are the future unless something more effective will be invented at some point.
I am aware that my rant sounds like "content is king" and this is true to some extent. If you disagree, try manipulating e-mail filter while sending shitty content. Then tell me whether content is king or not.
It doesn't mean that Google cannot be manipulated, though, through link building, social signals and other means. Just all these things are less important than content. Two websites of equal quality may compete with each other by building links and such. While a shitty website cannot compete with a cool website no matter how many links it shall have due to losing the user engagement battle.
