I know it is a long time ago but I think you can still be methodical about the search. He doesn't sound like the type of man to not stay in touch with his family if he was able to do so, so it seems something happened to him. Maybe he was in an accident and died; maybe he was in an accident and lost his memory.
Think about all the information you know, and try and put together a time frame and fact sheet that may help you search methodically, step by step.
Where did he work before he left, and what was his job? Was he laid off, or did he quit? Did he have a number of trades and skills? What were his hobbies?
Why did people think he went to South America? Did he have relatives there? Had others gone there from Cleveland at that time? Had there been some big economic development that would have atracted him? If not South America, was there somewhere else he was likely to have gone?
How would he have travelled to South America from Cleveland? Would he have gone to a port by train? What ships were travelling to South America in those times? What ports would ships have stopped at? Were there any shipping disasters?
Can you narrow down the time frame in which you think he left Cleveland?
Assuming he got to South America, either Argentina or Brazil seeming like reasonable possibilities, what churches would he have been likely to have attended?
What did his wife do when he disappeared? Did he own a house that was in his name? Was there any court actions to have him declared dead after a certain point in time, and if so, what kind of evidence was presented?
I am sure you can think of many more questions to ask. Some of them may not be relevant but it doesn't hurt to brainstorm all the possibilities. While records might be hard to find after all this time, you might be surprised at what it is possible to uncover.
This might seem like a lot of work but being systematic can often be very useful in trying to solve mysteries like this one. If you can determine the route along which he might have travelled, for example, you might be able to uncover possible accidents or deaths that might have occurred.