How does an Ubuntu upgrade avoid the crash?
Using ElasticSearch might help, but tuning MySQL is where I'd start. The links you posted didn't work for me the other day, but they do now. I just tried them and all work for me. The Lepidoptera page is noticeably slow, but not horribly so, to the point the I suspect tuning will resolve it.
If you can share your theme + data with me (or let me see the actual system) I can help you with this. We are already a vendor at the Africa Museum, so hopefully this is possible. Contact me at support@collectiveaccess.org if you wish to work directly on this.
Seth