Save the page when it has equity worth keeping and a fixable problem, and start over when the foundation itself is fundamentally wrong. The pivot is salvageable-equity-and-foundation versus fundamentally-wrong, and it turns on whether the page’s value is in what it has accumulated or whether that value is undermined by a base you would have to tear out anyway. Saving means updating or rewriting in place on the same URL. Starting over means building a new page and redirecting the old one to it. The decision is which costs less for the result you need, not which feels cleaner.
You save the page when it carries equity you do not want to lose and the problem is content you can fix. Equity is the accumulated value: the backlinks pointing at it, the ranking history it has built, a URL and structure that are right for the query. If all of that is sound and the only trouble is that the content is thin, outdated, or off-intent, you fix the content and keep everything the page has earned. Throwing the page away then discards real value to solve a problem you could have solved in place.
You start over when the foundation, the structure, or the intent is fundamentally wrong, such that fixing would cost more than rebuilding. If the URL targets the wrong thing, the structure fights the query, or the page was built for an intent it can never serve, patching it means fighting the base at every step, and you end up rebuilding anyway under the guise of a fix. In that case a clean new page, with the old one 301’d to preserve what equity it had, is both cheaper and better. The deciding question is whether the bones are sound and the problem is surface, or whether the bones are the problem.
To decide, inventory the page’s equity and locate the real problem. If it has links, history, and a sound URL and foundation, and the issue is fixable content, save it and rewrite in place. If the foundation, structure, or intent is fundamentally wrong so that fixing costs more than rebuilding, start over and 301 the old page. Save equity with a fixable problem, rebuild what is fundamentally wrong.