How can I make angle brackets function like lists?

The Notation package is built for this.

Needs["Notation`"];
Notation[ParsedBoxWrapper[RowBox[{"\[LeftAngleBracket]", "x___", "\[RightAngleBracket]"}]] \[DoubleLongLeftRightArrow] 
ParsedBoxWrapper[RowBox[{"{", "x___", "}"}]]]

(It looks better when input via the Notation palette. Don't be frightened by the box manipulation - I used the palette to construct that expression.)

If you want to see exactly how it looks in the frontend, a cell which contains the expression above looks like:

Cell[BoxData[
 RowBox[{"Notation", "[", 
  RowBox[{
   TemplateBox[{RowBox[{"\[LeftAngleBracket]", "x___", "\[RightAngleBracket]"}]},
    "NotationTemplateTag"], " ", "\[DoubleLongLeftRightArrow]", " ", 
    TemplateBox[{RowBox[{"{", "x___", "}"}]},
    "NotationTemplateTag"]}], "]"}]], "Input"]