This is not a criticism of web designers. Most are good at what they do, and what they do is build websites. DNS is a different job with very different consequences when it goes wrong.
The mismatch
A website going down is visible, annoying and fixable. DNS going wrong takes your email with it, fails silently in ways nobody attributes correctly for weeks, and is the single point on which your logins, invoicing and third-party tools all depend.
Those are not the same risk profile, and they should not automatically sit with the same supplier just because a website was the first thing you needed.
Three practical problems
Access ends when the relationship does. If a designer closes, moves on, or you simply stop working together, your DNS is somewhere you may struggle to reach.
Changes get made without full visibility. The classic failure is a domain repointed at a new host with the MX records cleared in the same edit. The site works; the email stops.
Out-of-hours support is rarely part of the arrangement. A broken DNS record on a Saturday is a business problem, not a design problem.
Where the line should sit
Your designer should absolutely be able to request DNS changes, and a good arrangement makes that fast and frictionless. What they should not be is the only party who can make them, or the registrant of record on the domain itself.
Most designers we work alongside are relieved by this. Holding a client’s DNS is a liability they never asked for.
See managed DNS and who owns your domain name. Call 01202 835235.