Why does .class:last-of-type not work as I expect?

Your issue is that you’re reading :last-of-type and thinking it works as a :last-of-class selector, when instead it specifically means elements only. There is no selector for the last instance of a class, unfortunately.

From the W3C:

The :last-of-type pseudo-class represents an element that is the last sibling of its type.

You have p.visible:last-of-type as your selector, which does the following:

1. looks at each list of elements (e.g. 1 or more elements; a child with no siblings can still have :last-of-type applied to it) within each containing element in your HTML
2. finds the last element in that list
3. checks to see if it is a <p> element
4. checks to see if it has the class .visible on it.

In short, your selector will only apply its styles to a <p> that also has the class .visible on it. In your markup, only the first two <p> elements have that class; the third does not.

Here’s a demo of different styles to illustrate:

p:last-of-type {
  /* this will be applied in the third paragraph because the pseudo-selector checks for nodes only */
  color: green;
}
p.visible:last-of-type {
  /* this does not get applied, because you are using the pseudo-selector with a specific class in addition to the node type. */
  color: red;
}

<p class="visible">First paragraph.</p>
<p class="visible">Second paragraph.</p>
<p>Third paragraph.</p>

Per your ultimate goal,

How can I select the last element inside the div WITH the class .visible? I do NOT want to use JavaScript for this.

The simplest and most performant way is to invert the way you’re trying to apply the styles; instead of trying to hide two out of three divs, where one of the divs to hide has a class and the other div to hide has no class, and the div you want to show shares the same class as the one div you want to hide which also has a class (see? that’s pretty confusing), do the following:

• Only add the class to the element (or group of elements) that’s smaller.
• Set the default style for the element to be what you don’t want the class to achieve.
• Set the style on the class to be what you want to achieve.

p {
    display: none;
}
.visible {
    display: block;
}

<div>
    <p>This should be hidden</p>
    <p class="visible">This should be displayed</p>
    <p>This should be hidden</p>
</div>

As you can see from this demo, not only are your HTML and CSS simpler, but this also has the benefit of using only a class selector rather than a *-of-type pseudo-selector, which will make the page load faster (see more on that below).

Why is there no followed by or parent selector?
This could potentially bog down the speed of a lot of webpages by changing just one class name dynamically on the page.

Dave Hyatt, while working on the WebKit implementation in 2008, mentioned some reasons why these implementations are avoided:

With parent selectors it becomes extremely easy to accidentally cause
a document-wide grovel. People can and will misuse this selector.
Supporting it is giving people a whole lot of rope to hang themselves
with.

The sad truth about CSS3 selectors is that they really shouldn’t be
used at all if you care about page performance. Decorating your markup
with classes and ids and matching purely on those while avoiding all
uses of sibling, descendant and child selectors will actually make a
page perform significantly better in all browsers.