The other guy was wrong as I suspected, so I'm going to answer my own question.
In SQL Server 2012 (and probably earlier versions), I found that if I call 'set showplan_xml on;' on an arbitrary query like:
select * from (select * from Lessons a) a inner join (select * from LessonTypes b) b on a.LessonTypeID = b.ID;
The returned XML includes the following as its first/top-level OutputList, which clearly shows all columns, and not only their source tables but the alias of their source subquery.
<OutputList>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="ID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="Name"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="Description"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="Enabled"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="LessonTypeID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[LessonTypes]" Alias="[b]" Column="ID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[LessonTypes]" Alias="[b]" Column="Name"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[LessonTypes]" Alias="[b]" Column="Description"/>
</OutputList>
So, in my database API, since I store all queries strings neatly in a dictionary, I can actually perform a cache warm-up at application start that runs them all through with xml_showplan on, parse the XMLs top-level output columns, and then have a complete mapping of table alias and column names to their output ordinal.
Actually, I've used a little trick here. It's not actually the alias of the subquery, but the alias of the base table. If you do not include an alias on the base table, then the Alias attribute is not present in the XML. However, it is very simple to make sure that every time you reference an actual table, you give it an alias, that should cause it to be output to the XML, and there you go.
Just to drive the point home, the aliases stick, even when joining the table to itself in a query such as:
select * from Lessons a, Lessons b, Lessons c
, which produces:
<OutputList>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="ID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="Name"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="Description"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="Enabled"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[a]" Column="LessonTypeID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[b]" Column="ID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[b]" Column="Name"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[b]" Column="Description"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[b]" Column="Enabled"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[b]" Column="LessonTypeID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[c]" Column="ID"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[c]" Column="Name"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[c]" Column="Description"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[c]" Column="Enabled"/>
<ColumnReference Database="[sensitive]" Schema="[dbo]" Table="[Lessons]" Alias="[c]" Column="LessonTypeID"/>
</OutputList>
So as you can see, the aliases are in fact intact and retrievable.