Just to be clear, do you mean that you set the site start page to uncacheable, but the problem still happens and you have to clear the cache to fix it?
Yes, performance will be worse if you turn caching off completely -- but that might be better than the site not appearing at all!
Is the homepage fairly static? If so, you might really consider the workaround I originally suggested until a better fix can be found. At least that would stop the client from calling you several times a day.
Simply copy the HTML source code for the homepage to a text file, and call it index-static.html.
Then add this line to your .htaccess right before the friendly URL redirect
RewriteRule ^$ index-static.html