Speed is not a technical vanity metric. A visitor arriving from a search result on mobile data has a short and unforgiving patience budget, and a site that makes them wait loses them before it has said anything. Worse, they rarely tell you. They simply go back and click the next result.
The first cause is almost always images. A photo straight from a phone camera is several megabytes; the same photo, resized to the space it actually occupies and served in a modern format, is a few dozen kilobytes. Most slow sites are one folder of unoptimised images away from being fast ones.
The second is plugins and third-party scripts. Every chat widget, analytics tag, tracking pixel, and social embed is code from someone else that your visitor has to download before your page is usable. They accumulate quietly over years. Auditing them is dull and it is usually the single highest-value hour anyone will spend on the site.
The third is layout shift: the page appears, then jumps as an image or a font loads late and shoves everything down. It is not slowness exactly, but it feels worse than slowness, because it makes people mis-tap. Reserving space for anything that loads late costs nothing and removes it entirely.
The fourth is hosting. Cheap shared hosting is cheap because your site is sharing a machine with hundreds of others. It is usually fine until the week you most need it not to be.
None of these require a redesign. If your site looks right and simply feels slow, the fix is nearly always maintenance rather than a rebuild, and any studio that jumps straight to "you need a new site" should be asked to show you the measurements first.