It turns out that calling a function with <<angle brackets>> always gives only the first output tiddler produced by the filter (unlike elsewhere in TiddlyWiki, this is true even if it's placed directly within wikitext rather than as the value of an attribute).
The most straightforward way to get the code snippet working as desired is by calling the function using the function[] operator (or the shortcut syntax) instead:
\function tiddlers-starting-with-C() [all[shadows+tiddlers]prefix[C]]
<$list filter="[function[tiddlers-starting-with-C]]">
<$link /><br>
</$list>
CamelCase
Changes
ChapterOutlineTemplate
Choosing a Type of Transclusion
Classifying Tags
Comments
Common Filter Operators
CommonTransclusionSyntax
Concept
Conditional Expressions
Contact Tiddlers
Copyright
Creating a List of Links and Backlinks
Creating Evergreen Notes
Creating Hierarchies with a Table of Contents
Creating Plugins
Creating Tiddlers With Predefined Fields
Custom Widgets
Interestingly, you can also fix this by making tiddlers-starting-with-C a procedure instead of a function:
\procedure tiddlers-starting-with-C() [all[shadows+tiddlers]prefix[C]]
<$list filter=<<tiddlers-starting-with-C>>>
<$link /><br>
</$list>
CamelCase
Changes
ChapterOutlineTemplate
Choosing a Type of Transclusion
Classifying Tags
Comments
Common Filter Operators
CommonTransclusionSyntax
Concept
Conditional Expressions
Contact Tiddlers
Copyright
Creating a List of Links and Backlinks
Creating Evergreen Notes
Creating Hierarchies with a Table of Contents
Creating Plugins
Creating Tiddlers With Predefined Fields
Custom Widgets
Puzzling through why this is may be enlightening. Spend a couple minutes seeing if you can figure it out before continuing! Consider what TiddlyWiki is doing as it evaluates a procedure vs. a function.
Answer
When tiddlers-starting-with-C is defined as a function, TiddlyWiki evaluates the filter in its body as part of the transclusion <<tiddlers-starting-with-C>> itself, because a function call causes the filter inside the function to be executed and output tiddlers to be produced. Then it passes those output tiddlers (or rather, output tiddler, since the angle bracket syntax always results in a single output tiddler) to the filter parameter of the $list widget. A single tiddler name is a valid filter, so the body of the $list widget is rendered once with that tiddler name as the current tiddler.
In contrast, when tiddlers-starting-with-C is defined as a procedure, TiddlyWiki evaluates the filter as wikitext, which simply yields the text of the filter itself (since there aren't any formatting instructions, transclusions, widgets, etc. in it). Then it passes this filter text to the filter parameter of the $list widget. The $list widget actually runs the text of a filter passed to the filter parameter, so this is perfectly valid, and completely equivalent to just:
<$list filter="[all[shadows+tiddlers]prefix[C]]">
<$link /><br>
</$list>