Your blog or website software should really be setting up those redirects for page moves automatically.
> You might be wondering why not just use Levenshtein's distance to find the page that the user was looking for. The reason is that Levenshtein's distance is only going to be useful in case of typos. Whereas vector embeddings will go further and find the pages that are the most semantically similar
Which is not at all necessarily what they wanted, compared to an honest 404! This would be particularly bad given that you fail to set up redirects for ordinary page moves/renames: this means that I could link to your post on 404 pages, you move it, the embedding fail and silently send all future visitors to a different post (eg. maybe you write a second post on 404 pages; after all, there's now a roughly 50-50 chance of which one would be ever so slightly more similar embedding-wise). This further means that anyone who updates their links to follow redirects will unwittingly bake the lie into their page, making what used to be a working link or an honest error into a false live link.
In general, I would suggest that you set up correct redirects to obviate most of it, use embedding search to curate possible redirects, and only present a list of suggestions on a genuine 404 error.
(Incidentally, if you have a good list of redirects, Levenshtein search on the list itself with a new error will often correctly generate a redirect. So you don't even need embeddings/retrieval to fix a lot of regular 404s. I'd say the majority of my remaining 404s are handled by my Levenshtein search script.)