Which is a closest metric that determines the website quality/functionality? Whether it is Visitors, Page views, traffic sources, Conversions, Events, E-commerce purchases or any other parameter?

there are so many things present on analytic report from which you can measure the effectiveness of each thing on your site. For example if you are uploading article on your site and if you want to know how much people actually reads your article, you can have a look at average time on site section of that specific page and compare with the average time needed to read an entire article.

You can't just stamp one technique as an important metric.

Its actually important to track each of these metrics.

Example being you need to know from where you are getting conversions and for that you need to know the track the traffic sources.

You need to track how many people came to your website and just left so that you can work upon you areas which are increasing your bounce rate and so on..