I have used it on production sites for years, and in fact, have not used mod_php since before PHP5 for the exact reasons you describe.
You may need to tweak your FCGI settings a bit - the amount of children/fcgi process lifetime, etc all make a difference.
One thing I would highly recommend is ensuring you run under a PHP accelerator (APC is free/great), particularly with FCGI. This will reduce the time each PHP-FCGI child takes serving a PHP request (as it doesn't need to reparse the code every time), which means you can have less children. Less children means more memory available and we all love memory.
If you're having specific stability problems with fcgi, do some googling on the errors you're getting. There definitely have been bugs in the PHP fcgi interface, so an upgrade to later/latest version may help you there.
Alternatively, something like lighttpd/nginx could be a good option for you (though they both recommend fcgi if you want them talking direct to PHP as well).
Good luck!