There is no easy answer to this question. It depends entirely on the objectives of the site at large and of each page.

IMO, the best way to answer the question is to write the pages as they most make sense to the user. It's a cliché, but for good reason. The SE's like natural language pages and are getting better and better at sussing them out over time, and that should continue. I would not go around stuffing kw's into the pages for the sake of getting the kw's into the pages, unless doing so conservatively doesn't look at all odd or strange, and is either a neutral or positive change to each page's intended communication to users.

FYI the SE's have filters that account for such things, and over abundance of a single kw phrase sitewide can actually work against a site in some cases, especially earlier in the site's history.

Yes, I do believe the presence of atleast one theme keyword in each of the pages, intended to be promoted. This would give you an edge to promote your website (its multiple pages) on broad based keyword spectrum.

I have numerous very successful sites where the core kw phrase is not present in the body of every page, or even in every page title. That is intentional, and an important contributing factor to those sites' success.

OTOH, I have more sites where the core kw probably is found on every page -- or close to it.

It really does depend upon lots of factors.

"It really does depend upon lots of factors."

Very true, placing one or not placing any theme keyword at all depends on your intended objective (what you would want to achieve by this).