This whole business of trusting God is completely alien to me.
Let's look at the scenario posed in the OP. What does trusting God look like in this case? Is the act of trusting God with the safety of one's family expressed in removing all one's door locks, fire alarms, phones, etc? No one does this. Even the most fervent believer, walking in the full expectation of miracles from God each and every day, does not act insensibly as an expression of this "trust". Believers do not picket hospitals and police departments out of a righteous desire to remove these temptations of trusting that which is not God. There is, in this context, no case of trusting God.
Accordingly, therefore, it is meaningless to fear that acting sensibly means you don't trust God.
So what is the meat of the problem here? What is really going on? There are two problems with this proposition - what we mean by trust, and what we mean by God.
You cannot say that you trust something without the suggestion that it is possible for that trust to be betrayed. For example, you never hear anyone talk about trusting the sun to shine, because it is inconceivable that it will fail to shine. Is it possible for God to betray you?
The second problem is one of scope. It is imagined that there is a line drawn across the universe such that God is only on one side of it. In the case of this discussion, that which is considered to be on the other side is sensible action to protect one's family. This is nonsense. The reason we don't picket hospitals and police stations is because we know that God is in those places, too. God's power pervades all things. All.
How is it that we have arrived at such incredible misconceptions that there are places God cannot go and things with which he cannot be "trusted"? These ideas are completely incompatible with the nature of God. Where did we get them? We confused the nature with the concept. If you say the word "God" out loud (do it, right now), it is just a sound. If I write it here, it is simply an arrangement of electrons on your computer screen. If I write the most beautiful piece of prose about the incredible magnitude of God, those words cannot approach that which they represent. I could craft the most gloriously open-ended concepts anyone has ever read, and if you took my words to be the nature of God rather that the concept, we will both get it wrong. We will wind up with a God in a cage, like a lifeless child's toy, which we will begin to speak of trusting because we never actually see it do anything.
This is idolatry. To confuse something which represents God for the actual surpassing, timeless, infinite presence of God is to worship an idol. We know that idols can be physical objects. But the worst objects of all are objects of the mind - concepts and words.
God is not that which is imagined. He is unimaginable. God is not that of which we speak. There are no words. God is not what we think he is, for he is beyond our thoughts.
