Live link not working as expected

I’m not seeing anything specific about a DNS error in either of those logs. Do you have the specific error that you are encountering?

One thing I did notice is that this site appears to be using Elementor.

It looks like this site uses the Elementor page builder to create pages. This is great, but one issue that can happen is that the way that Elementor stores the styling data makes it hard to update the URL when accessing the site over a Live Link. Specifically, Elementor stores the styles for individual pages in external CSS files which Local can’t edit to update that url.

What you need to do is log into the WordPress admin and navigate to Elementor’s settings page under “Elementor > Settings > Advanced.”

On that page, you’ll need to change the “CSS Print Method” to “Internal Embedding”. This will make is so the Elementor styles will not be written to files, and therefore are using the correct URLs that are found within the database.

See this related topic for more details about that setting: