Wolfman I checked your link, and before I even read your post, I can see why it's loading incredibly slow. You have a combination of settings that is forcing the ORIGINAL image to load as small thumbnails. See screenshot:

Your images are massive (up to 15 MB as you can see in the screenshot), and just to load a few thumbnails, I had to load 2-300 MB into browser. So not only is this extremely slow, but it will require a massive amount of memory in browser, and the page/scroll will become unresponsive. If I was to start scrolling this page, it would "download" 1.7 GB into browser, just to display the thumbnails. I'm not sure what you are trying to do here, but this is wrong on so many levels ...
Wolfman But it takes a really long time for my clients to see the images and be able to download
Yes, because of the above, it will obviously be incredibly slow. As you can see, a massive difference from our demo.files.gallery.
I'm not quite sure what you mean "and make a PDF" ...
Wolfman Because of the slow response of the image grid, my method is as follows: I do the upload, then log in, scroll down and wait for each image to be rendered, hoping this process will be faster for my clients afterwards.
It won't be faster. You have literally disabled image resizing, so it's forced to load extremely large original images into the grid. That's why it's slow.
Wolfman /_files/cache/images is empty and I don't know why. Maybe a misconfiguration?
You have a couple of settings that you have customized
// here you have reduced from default 60000000 to 10000000
// basically you are telling Files Gallery to not resize large image (all your images are large)
'image_resize_max_pixels' => 10000000,
// here you have increase the value from 1 MB to 20 MB
// basically you are telling Files Gallery "it's ok to load extremely large images as thumbnails"
'load_images_max_filesize' => 20000000,
From the above, you have basically disabled image resizing, and enforced loading extremely large images directly into the interface. As noted above, this will be extremely slow, as the point of resizing images + cache is entirely removed.
Of course, in your case, you have extremely large images, that will take quite a lot of power to resize. However, this is the only way to speed up performance of loading images, by serving much smaller resized images.
Wolfman Three images are not visible in the grid. You see the green jpeg placeholder image. They're really big and I think the server doesn't want to render the previews. Any solution?
Because your images are extremely large, and they exceed the 20 MB limit (which is already way too much). You need to server resized images to speed up everything.
Wolfman Maybe a URL call with "?action=render-cache" could start the rendering cache process.
If you have 2GB worth of extremely large images, you need to understand what you are asking here. This process will likely take more than a day on your server, while at the same time, the server is locked in a process of using 100% CPU, which will likely make your server extremely unresponsive for all other tasks.
Wolfman This opens up another avenue for me: I could render the previews locally into the same folder as the original images, taking into account the necessary filenames, and include them in the upload. The upload will take longer, but this would be the fastest way as my local machine is much faster than the server rendering, and uploading is usually quite fast with 5G or wifi.
You mean create resized image cache locally on some "desktop", and then upload to server? With the new cache system coming soon, this could be possible, but would only serve purpose if your "local" desktop is much more powerful than your "server". Yes you would need to upload the resize cache, but these files are rather small (10-30kb) compared to originals (2-20MB).
In either case, I think you need to understand that there is no "magic wand" solution here. You simply can't resize that amount of large images any faster than your server can process them. And as you can see, serving the original images (instead of resized cache) is really counter-productive also.
I think your first task however, is to revert some of your config options so that you actually start loading RESIZED images instead of the originals. Then see how fast/slow it is (it probably won't be slower than loading the full originals). One thing is for sure, it will definitely be much faster as for any visitors as soon as the cached images are already created.
Your case is a bit special. There really isn't a simple scenario to create and display small preview images swiftly, without going through a tedious "resize" process.