All you'll be doing in a history, theory, and criticism degree  is reading and writing.  The admissions committee wants applicants who can already perform these decently already.  So, yes, they want to see something that you've written, and written well.

To maximize your chances, pay the same level of detail to this paper that you would your portfolio for a design program application.  Proof it, have someone smart read it to make sure it's clear and error free, proof it again, and revise as necessary.  This includes your footnotes, bibliography, and formatting.  Words and writing are the basic tools used in an HTC program, so make sure the sample you submit is essentially perfect.

Also, in the absence of more information, a 5-page paper may be as good as a 25-page one, if its content demonstrates your skills adequately.

But calling the school and asking them is always the best strategy.

Good luck!

just discovered this tucked away on the website (still a little puzzling since, as you say, students would only be reading and writing in the HTC program!) so perhaps i'll include the writing sample as part of the portfolio.

Portfolio: Required for MArch, SMArchS, SMACT. Portfolios are optional, but recommended, for SMArchS HTC, SMBT, and PhD. Writing samples may be substituted where relevant and are expected for HTC PhD. Portfolios will be digital PDF files. For most programs applicants will upload their files to the online system. ACT applicants will mail their portfolios on CDs.