Why can't I reorder my SQL Server columns?

Definitely you can. Uncheck this option in SQL Server Management Studio:
Tools > Options > Designers > Prevent saving changes that require table re-creation.

Please don't do this in production unless you know the implications!

Reordering columns is basically the same as remaking the table, which is the workaround if changes that require recreation are disabled:

SELECT (fields in DESIRED order)
INTO MyNewTable
FROM OldTable

You might be interested in the answers to this question.

As to why you can't reorder columns (aside from the SSMS-specific answers posted), I'm not sure you'll find a canonical answer, other than the SQL standard contains no programmatic syntax for doing so - the only way to redefine a table's schema (including the order of its columns) is to drop the table and recreate it. So any management application that provides column "reordering" will most likely be doing that behind the scenes (e.g. SSMS's Design View, as pointed out in the linked question).

Someone with deeper knowledge of the SQL specification could contradict this if there is an explicit reason for not providing a column-reordering mechanism as part of the standard DDL, but I can only conjecture that, since queries define the order of columns in their resultsets anyway, and DBMSes control physical storage according to their individual implementation, that logical column order is essentially meaningless and doesn't warrant such a mechanism.