Sitewide linking to one page helps until the link stops looking like a genuine recommendation and starts looking like a manufactured pattern, and the line falls at relevance plus placement, not the mere fact of appearing everywhere.
Here is the line said plainly. A link that appears on every page because it belongs in the site’s navigation (your main service page, a key category, the contact page) reads as legitimate structure, because users genuinely need to reach those pages from anywhere. A link that appears on every page because someone wanted to force value into it (an ordinary blog post jammed into the footer of all 400 pages it has no contextual reason to sit on) reads as a footprint, because there is no user reason for it to be there.
What tips a search engine toward seeing a pattern is the mismatch between the link’s ubiquity and its relevance. Navigation links are ubiquitous and relevant, so they pass as structure. A non-navigational page linked from everywhere is ubiquitous without being relevant to most of those pages, and that gap is the signal. The links also tend to share identical anchor text and identical placement, which compounds the manufactured look.
The practical effect is that sitewide links to genuine navigational destinations are fine and expected, while sitewide links engineered to pump authority into an ordinary page are discounted at best and a risk at worst, because the value gets spread thin and the pattern reads as optimization rather than editorial choice.
Test your own sitewide links against the line: for each page linked from everywhere, ask whether a user landing on a random page would actually need to reach it. If yes, it has a reason to be in the template. If the only reason it is there is to lift its ranking, you are on the wrong side of the line, and concentrating that link where it is genuinely relevant will serve the page better than stamping it everywhere.