They'll be infinitesimal faster for the server which will sit idle most of the time anyway... and much slower for you when you make a minor header/footer mod locally, and then you need to re-upload all pages that have a header (i.e. all pages).
I'm sorry, but this project feels more like "art" than a practical tool.
Plus, with a few basic lines of PHP (available on every shared host) you can pre-generate your site statically as well (and on the server, to save you the re-uploading).
Presumably the 're-upload all pages' bit can be automated. As far the reset, the trade-off depends on how often you're updating content versus the amount of traffic coming to your site; in many cases, a tiny saving on frequently-occurring page delivery is well worth a big delay on one-off page generation.
http://httpd.apache.org/docs/2.2/mod/mod_disk_cache.html
re:edit hxincl uses a similar style as ssi but with a couple of enhancements:
http://manpages.ubuntu.com/manpages/trusty/en/man1/hxincl.1....
if you want exactly SSI syntax, ain't nuthin' stopping you from pointing curl at localhost.