Confirming this. I have issues with the oembed links in the header becoming 404 since they only work with the original permalink and not with the custom permalinks as they are considered redirects. Not sure if this is even possible to fix from your end.
<link rel="alternate" type="application/json+oembed" href="https://website.com/wp-json/oembed/1.0/embed?url=https%3A%2F%2Fwebsite.com/new-slug" />
<link rel="alternate" type="text/xml+oembed" href="https://website.com/wp-json/oembed/1.0/embed?url=https%3A%2F%2Fwebsite.com/new-slug&format=xml" />
The above will not work but the use with the old names does.
<link rel="alternate" type="application/json+oembed" href="https://website.com/wp-json/oembed/1.0/embed?url=https%3A%2F%2Fwebsite.com/old-slug" />
<link rel="alternate" type="text/xml+oembed" href="https://website.com/wp-json/oembed/1.0/embed?url=https%3A%2F%2Fwebsite.com/old-slug&format=xml" />