A records are usually fine for web dev, and some cname records. But if you move DNS to different name servers it doesn’t take the MX, srv, or txt with you.
Name servers are where all the DNS records are hosted. It tells every computer who is the authority on the information. If you change those without moving all the DNS records to the new location. All the old ones are no longer used at all. Even if they still exist on the old system.
Also spf records might also reference the websites a record to allow the website to send email without getting flagged as spam. If the site moves, but emails are still sent from the old IP address for a myriad of reasons it would break email. Like if they used some provider that hosted the site and emails on the same system, which used to be more common than it is today.