Grist does have an API that lets you upload attachments. A script could go through the list of URLs, download each one individually, and then upload each one to Grist via the API. I don’t think anyone has written and shared such a script yet though, so it would require some effort and knowledge of programming.
Does Airtable provide any way for you to bulk export all of your attachments as files (and not URLs)? If yes, the link from your original post has a workaround on how you might be able to bulk import the attachments into Grist.
George