Well, there's "on-page SEO" and "on-page SEO."
That one makes a specific on-page change or changes and sees no ranking boost may speak to the efficacy of specific techniques, or on-page structural signals, but not to the general task of "on-page SEO."
Structuring page content to make it as understandable as possible to the search engines is a task that will always bear dividends in the end, and to that degree is always a necessary part of SEO.
Following a bit on tedster's point, Google may no longer "reward" you to the same degree it once did for having "blue widgets" in an H1, but all things being equal your page will rank better for "blue widgets" if it contains the compared to an identical pages that does not. That picture of a "blue widget" is much more likely to have visibility in image search if it's captioned "blue widget" and contains "blue widget" in the img alt attribute.
Whether or not one frames this as "relevance" or "on-page SEO that influences ranking" the search engines are continually sifting through the semantic universe of individual documents and trying to make sense of them. To that degree, at least, "on-page SEO" will always have a place in helping the search engines extract classifiable meaning from words on a web page.