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This isn't an answer to the original question, but rather explains why I don't use the “Equal Symbol Aligned Math Cell”. Note that I used to use the writing palette to create aligned math, but now I use the solution presented below. And I have some really large notebooks full of typeset maths/physics using this solution.

The Writing Assistant palette contains the button to create an “Equal Symbol Aligned Math Cell” which is actually a GridBox embedded inside of a "Text" cell. Specifically, it creates

Cell[TextData[Cell[BoxData[
    FormBox[GridBox[{{GridBox[{{"\[Placeholder]"}}, 
         GridBoxAlignment -> {"Columns" -> {{"="}}}]}}, 
      GridBoxItemSize -> {"Columns" -> {{Scaled[0.96]}}}], 
     TraditionalForm]]]], "Text"]

I find this a little hacky and (as mentioned in the comments to the question) a bit unstable.

Luckily, Mathematica has a simpler alignment system built-in. Just insert a DisplayFormula cell and set the cell alignment from Left to AlignmentMarker using the Format ► Text Alignment ► On AlignmentMarker. Then you can set the formula to align where ever you want, using \[AlignmentMarker\], which can be entered using ⁝am⁝

Cell[BoxData[{
  RowBox[{"a", "\[AlignmentMarker]", "=", "b"}], "\[IndentingNewLine]", 
  RowBox[{RowBox[{"c", "+", "d"}], "\[AlignmentMarker]", "=", "e"}]}], 
 "DisplayFormula", TextAlignment->AlignmentMarker]

which in TraditionalForm looks like:

Then, if you want, you can make a new cell style that uses this option (or modify an existing style) and assign that style a MenuCommandKey as described in here or in Adam's answer to this question.