Earlier quoted context omitted.
Factory reset and wipe your device before travel, and restore when you get to your hotel. I’m a privacy nut and while this is a disturbing trend there exist straightforward work arounds. EDIT: I do this all the time and it’s not even remotely difficult or a big deal.
And how do you restore the phone? I assume you travel with a laptop? What happens when they want access to the laptop?
Login to a cloud server that has your backups / restores / setup scripts / data. Preferably encrypt those backups before you upload them to the cloud server.
Only reason why I don't do it more frequently is it's a pain in the ass. I've been fairly impressed with iOS's backup restoration system recently.
Only had to log in to a specific set of services for everything to be back to normal.
This system is definitely a great way to test your backup restores!
Also: Don't use email for any semblance of secure communications.