Automatic Google account backups make this useless
Instead of torturing you to get them to give up your phone password, they torture you to give up your Google account password. Same applies to an iPhone with an iCloud password.
Sure, you could have a second innocuous account you give them, but if they know you have a known association through other means (such as having already gotten into someone else's call/text history and know you've contacted them) they'll see through this quickly and torture you even more.
You simply can't have any backups if you want to adopt this strategy. Surely if you're in a repressive regime losing your contact list is the least of your problems if you're taken in by the authorities? Though perhaps better than wiping your phone would be a controlled wipe that deletes certain information but leaves other information intact. A freshly factory reset phone is going to be treated with suspicion by the type of people who consider rubber hoses a warmup for the real interrogation.
While alerting your family/friends what happened to you is a very good idea, using SMS for that alert is a terrible idea. Repressive regimes tend to own or control the telecom infrastructure, so that text message just gives authorities a way to find out who the important people are in your life and their phone number. Good luck trying to give them a fake name if your phone alerts your mother, your two sisters and your wife. Better to send some sort of encrypted message through a service that isn't under the control of the regime, and where the list of addressees can't be seen by the authorities.
If you must use SMS, it should be to some out of country number that receives a code word/phrase that is automatically translated by the computer receiving it into a separate SMS to those you designate to prevent the government from identifying them quite so easily. And only works once, so the authorities can't keep texting it and figuring out which numbers in their country receive messages every time they do.