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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. Topological scaling (?)

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. How to post a file (This answer contains only a link to imgur.com, not to a specific picture.)