What you have expressed above, Dave, is very close to my own view. The problem of evil has been debated by philosophers and theologians for centuries, and no one yet has come up with a complete answer. I do think that there are partial answers which have been offered:
God does little or nothing to prevent:
1. Suffering From Other People
in order not to interfere with the free will of man. He wants all people to relate to Him of their own free will.
2. Suffering From Natural Disasters
a) in order not to disturb the workings of fallen nature. Particular animals sometimes kill and even eat people.
b) in order not to disturb the workings of nature as God created it. For example if when people stumble and fall over a cliff, if He caused them to land like a feather at the bottom of the cliff, there would be no uniformity with the usual working of the force of gravity.
I know there are many questions, even about the above explanations. And these two are probably not exhaustive in explaining why God “allows” some forms of suffering.