Redesigning a website can be a pain if not planned well. One – you’re dealing with a huge beast that already has several moving parts. Two, search engines already have crawled it well, and not to keep it all in tact while redesigning!
SEO can be a problem if proper planning and testing is not done while redesigning a website. Because, even a small change in navigation can create catastrophic problems in the site’s health.
SEO should always be given top priority while redesigning website. The reason is that, SEOs will be able to predict the effects of a design change and if you’re a business that relies on organic traffic, you have to prioritize traffic to aesthetic changes.
Having said that, design should be also focused on UX and the best ones are usually a good balance between SEO and UX.
I found this resourceful article about things to consider while redesigning your website and how SEO plays a major role in prioritizing things while doing the redesign.
It talks about how you should be planning your website redesign with SEO in mind.