Whether an omitted "that" should be replaced with a comma in certain situations

As with all punctuation, you have to ask not so much whether or not it's correct as what purpose a comma here might serve. With or without that, it wouldn't seem to serve any. On the contrary, it's a distraction and your intution is right.

No comma. When determining where the commas go, parse the sentence as if all the implied words were present. You can remove "that" because it is implied, but the meaning and function of "that" is still present.

Here's another example of comma use with an implied "that": "He discovered he was hungry and the fridge was empty." You might think you need a comma after "hungry" because this example seems to have two independent clauses connected by a coordinating conjunction. If this sentence is communicating two independent ideas, you would be right to add the comma. However, if this sentence is about what he discovered, you would be wrong to add the comma. Let's look at the sentence again, this time with all the implied words in place.

"He discovered that he was hungry and that the fridge was empty." As you can see, this sentence is now about discovering two things. The subject and verb are "He discovered," resulting in one independent clause expressing one idea.

What's the point here? You need to punctuate the sentence as if the implied words were present. (Zen Comma rule E: Use commas as if implied words were present.) Your sample doesn't need commas with "that" in place, so it won't need commas when you imply "that."

I think that (like many punctuation errors), this happens because the writer isn't clear about exactly what is meant. The phrase without a comma would, as you say, be fine. You could, alternatively, insert a semicolon and slightly alter the meaning. If you're not sure which you mean, then obviously inserting a comma will be halfway between the two, right? Wrong.