When I worked at a church I did the exact same as you: briefed 2-3 folks on staff on what they might get asked while I was out (and I was out of the country over Christmas!), and what to do. Of course you can’t cover absolutely everything, but I tried.
I made a few folks an agent (for Helpdesk tickets) or an administrator for a few very specific tasks they might need access to while I was out. This took a good amount of extra work but it was worth it to be in the UK and not have any worries about either being contacted or issues that might arise that someone I delegated couldn’t handle. [It also helped that I put a different SIM card in my iPhone so the easy way to text me wasn’t available.]
Like you I asked to have a representative from an MSP on site while I was gone, and he was able to handle everything. I also left detailed contact sheets with all department heads at all our locations.
Thankfully, no big issues occurred and what did come up was handled by the folks empowered to help. Sounds like you’re on the right track.