I have created a short animation for testing purposes only. I installed ffmpeg so that I could export as an mp4. However, when I export, the pictures come out fine, but the mp4 file is a one second video consisting of all black frames.

What went wrong here, and how do I fix it?
Let me know if you need a log.

When you say 'export', I assume you mean you used File -> Render Animation. The 'pictures' are the sequence of .png files for each frame which Krita tells ffmpeg to use as input, so Krita seems to be working fine up to that point.

I thought I had ffmpeg properly installed on my system but the only way I could get an animation to render was to go to https://www.ffmpeg.org and download the static release build. Then I put that in a sensible location and entered its file path in the FFMpeg field of the Render Animation options window. After that, it started working. I suggest that you try that.

I use Render as GIF image for output because that gives perfect rendering in an animated .gif but with a large file size. If I used the video output formats I found they had high compression that ruined fine detail.

One possibility would be to learn how to manually drive ffmpeg from the command line and tell it the .png file sequence and the amount of compression, etc, so you can get the quality you want. That could be a steep learning curve and a lot of hard work though.

It might also be worth it to check if the same happens with ogg or mkv, and whether changing the profile makes a difference.