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Not an answer, but doesn't really fit in a comment.

Glorfindel ran a script to find and try to repair/replace linked images. His answer includes a link to a (very long) output of results. Searching for "mathoverflow" jumps to the results for MathOverflow (and also meta.mathoverflow). Many images were successfully replaced. There are only a handful of uncertainties or failures. Here they are (I "snipped" out the successful replacements):

## mathoverflow.net

### Successful replacements

[snip]

### Uncertainties

| Frequency | URL | Replacement | Post IDs |
|----------:|-----|-------------|----------|
| 1 | https://i.imgur.com/KFkbXCz.gif | Not uploaded but archived as https://web.archive.org/web/20230503101817/https://i.imgur.com/KFkbXCz.gif | 291829 |
| 1 | https://i.imgur.com/p7zUONx.gif | Not uploaded but archived as https://web.archive.org/web/20230503101846/https://i.imgur.com/p7zUONx.gif | 375339 |

### Failures

| Frequency | URL | Failure | Post IDs |
|----------:|-----|---------|----------|
| 1 | https://i.imgur.com/76yeO.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Invalid Content-Type application/octet-stream returned for https://web.archive.org/web/20150427162752///i.imgur.com/76yeO.png | 39637 |
| 1 | https://i.imgur.com/7ctju.gif | Not found in Wayback Machine. | 30136 |
| 1 | https://i.imgur.com/Gnpdn.png | Not found in Wayback Machine. | 25897 |
| 1 | https://i.imgur.com/RleJK.png | Not found in Wayback Machine. | 30136 |
| 1 | https://i.imgur.com/bj5zs.png | Not found in Wayback Machine. | 30136 |
| 1 | https://i.imgur.com/cZKaB.gif | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 24272 |
| 1 | https://i.imgur.com/pChFw.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 56138 |
| 1 | https://i.imgur.com/xAZZb.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 38236 |
| 1 | https://imgur.com/F1x6j.png | ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Not found in Wayback Machine. | 35780 |
| 1 | https://imgur.com/drlwRgv | DownloadFailedException: Image not found on imgur image page. | 168300 |
| 1 | https://imgur.com/xjxWV.gif | ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Not found in Wayback Machine. | 24272 |

## meta.mathoverflow.net

### Successful replacements

[snip]

### Failures

| Frequency | URL | Failure | Post IDs |
|----------:|-----|---------|----------|
| 1 | https://i.imgur.com/76yeO.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Invalid Content-Type application/octet-stream returned for https://web.archive.org/web/20150427162752///i.imgur.com/76yeO.png | 5551 |
| 1 | https://i.imgur.com/cZKaB.gif | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 5551 |
| 1 | https://i.imgur.com/pChFw.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 5551 |
| 1 | https://i.imgur.com/xAZZb.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 5551 |
| 1 | http://imgur.com/ | ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Invalid Content-Type text/html returned for https://web.archive.org/web/20230509011314/https://imgur.com/ | 773 |

Here is my attempt to make easier links to the MO and meta.MO posts (some of them have several broken/missing images, I just put one link for each post):

1. Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

2. Numerical instability using only Heun's method on a simple PDE

3. When to throw away data

4. What are good ways to 'relax' a uniform approximation into independent saddle-point expressions once the uniform approach is no longer needed?

5. Why does the area function of a parallelogram have a nonintuitive geometric solution?

6. https://mathoverflow.net/questions/35780

7. https://mathoverflow.net/questions/375339

   This gif has been replaced and it seems to be the correct one.

8. Symmetries of the Julia sets for $z^2+c$

9. How to formulate such problem mathematically? (line continuation search)

10. The hexagonal property of Pascal's triangle

11. What is the status of the imgur images on MathOverflow?

12. https://meta.mathoverflow.net/questions/773