How can I force the WordPress visual editor to preserve the line breaks after paragraphs?

The editor is doing fine to add <p> tag on line break. I think what you need is to add some css to add margin to <p> tag so you don’t have to add additional or empty paragraph tags (which i see on your current page).

try something like this on your css:

.entry-content p{
    margin: 10px 0;
}

That should give you a nice margin on top and bottom of the paragraph tag. Also you might want to add some margin on <ul> and <li> to get a good formatting.

Hope it helps. Let me know if you have any questions.