Fixing is usually quicker and cheaper up front. The risk is that you keep spending on a base that will need replacing within a year or two anyway. Ask for an honest view on how long the current site has left.
Replacing gives you a clean start, but it carries a different risk. If your page addresses change and the old ones are not redirected, Google can drop pages that used to bring in visitors. Google advises mapping every old address to its new one and using permanent redirects.
Google also says to expect some temporary movement in rankings during any site move. For a medium-sized site, it can take a few weeks or more for the new addresses to settle in search results.
If your current site already ranks well for important searches, don't switch it off and start again without a redirect plan. Moving pages without redirects is one of the most common ways small businesses lose search visibility after a new website.