After stumbling around with this problem for a few days, trying to get server side includes working, I came across this article that solved my problem. This person was having the issue under 10.2.6, and it apparently still applies under 10.3.4.
Specifically, what I did in the httpd.conf file was to scroll down to where the web directory was actually defined, and added in the 'Includes' option there. Of course this was also after uncommenting the two AddHandler lines mentioned in that article.
I now have something like this:
# This should be changed to whatever you set DocumentRoot to.
# This may also be "None", "All", or any combination of "Indexes",
# "Includes", "FollowSymLinks", "ExecCGI", or "MultiViews".
# Note that "MultiViews" must be named *explicitly* --- "Options All"
# doesn't give it to you.
Options Indexes FollowSymLinks MultiViews Includes
A test document with a .shtml extention worked as intended.