The "mark as fixed" feature in Google Search Console is merely a tracking mechanism to see how Google reprocesses issues, but it doesn't expedite the process itself.
Regarding 404 pages, John Mueller clarified that these should generally remain 404, as they indicate non-existence, and it's primarily for tracking purposes.
If there are issues with 404 pages, it is crucial to resolve them properly by implementing redirects, returning necessary content to 200 status, and checking internal links.
For newly designed sites, if the recognition of changes isn't rapid, it may take longer to be reprocessed, especially for substantial changes.
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