We discovered the cause of those periodic outages, where Apache would just stop running - turns out it was caused by the Cpanel app installed on that server. It was rotating the Apache logs so they didn’t get too large — and then doing a restart of Apache to force it to use the new log. Problem was that the “graceful” restart that was supposed to preserve connections — didn’t preserve the connections. It just dropped everything.
We’ve got it tweaked now so that won’t happen any more, and right now everything is really blazing along.