SUCCESS!! We have a nice site-wide performance boost!
More recent editions of PHP (since v5.5) incorporate a built-in post-compilation interpreter opcode caching facility. That facility is disabled by default, and it had been. It's now up and running, configured and tuned. I've been monitoring its performance counters and it appears to be working very nicely. Our pages feel snappier such that it no longer feels laggy, and I think this will allow the site to scale well.