It's nice for a website to look nice. But it's not that important.
What's important is whether your users can find, read and understand your content and complete actions easily.
That's it. Everything else is secondary.
Can your users locate information and services or is it a constant guessing game? Good findability includes SEO, clear labelling, high scent-of-information, usable navigation, etc.
Can visitors engage with ease or is information shovelled online in dense clumps? Good readability includes clear information flow, good layout and white space, suitable fonts, etc.
Can readers absorb your messages first-time or are they full of obscure verbiage. Good understandability relies on plain language, no jargon or legalese, etc.
Can users complete tasks easily or are they faced with awkward interactions and errors. Good actionability has clear controls, clear trajectory, clear micro-content, etc.
You probably already know all this.
However, I bet have you colleagues who talk a lot about the "look-and-feel" and want another "redesign" as a cure-all for endlessly disappointing performance.
They are stuck in the era of websites as art projects.
They don't get it. Websites are not art projects. They're engines. They're machines for doing things. Websites are supposed to be greasy and constantly tuned and tinkered with — not admired from afar.
That doesn't mean they have to be ugly or austere.
Most modern websites are built on responsive frameworks and have good aesthetics and other features (mobile support, accessibility) by default.
Absent some fundamental business change, there is no excuse why such a site should be rebuilt until every ounce of utility has been squeezed from it.
Imagine if you could just work on the website you've got, instead of being told to "redesign".
You could concentrate on what's important by making content easier to find, read, understand and action.
You could restructure your Information Architecture. Change labelling in navigation. Rewrite your content. Simplify interactions. All without a redesign.
The more websites I look at, the more this lesson becomes clear.
Great websites don't redesign - they just get better.