I finally gave up resisting all the peer pressure and installed WordPress on the site. As expected, the installation went just fine and the basic install worked OK right out of the box. There were a couple of things I needed to do though:
Converting the old blog posts to the new format was fairly simple – just read them all out from my existing database table and put them into the wp_posts table for WordPress to use. I wrote a script to do this bit and after a couple of false starts it went through the thousand-odd entries and moved them over.
The only real problem I had was getting the old blog URLs to work in the new format. For reasons lost to me now, I had decided to use only 2 digits for the year, whereas WordPress uses 4. A quick RewriteRule to affect a 301 redirect to the new URLs, and all my cached Google links are working again.
I better investigate some themes and plugins later.