Some wag of a copy editor once said that his entire career had consisted of replacing that with which and which with that. In my case, it has been more than three decades of inserting that when a reporter had omitted it or, worse, another copy editor had deleted it.
The blame for this (as with the idiotic "split verb" superstition) I lay at the feet of journalism schools and their predilection for non-idiomatic English.
That used as a conjunction, or subordinator, is often, and safely, omitted when it links two short clauses: I told you I was sick. This is common in conversation and unobjectionable in formal writing.
But there are many occasions on which using that is necessary and desirable, a circumstance that even the Associated Press Stylebook, for Fowler's sake, understands.
The AP Stylebook, for example, advises to use that when a time element follows the main verb. I wrote yesterday that the premier/premiere distinction should be maintained.
Also, several verbs idiomatically require the use of that: assert, contend, declare, estimate, make clear, point out, propose, and state. I contend that even journalists are capable of instruction.
And it is required before subordinate clauses that begin with after, although, because, before, in addition to, until, and while. You can see that though journalism students have been badly taught, reliable advice is available.
Further, if a main clause is followed by two or more subordinate clauses, each should be introduced by that to give the reader the benefit of parallel construction. I have showed you that the subordinator is often necessary and that the occasions are easily identifiable, even by a journalist.
The AP Stylebook's advice—God above, I'm reduced to endorsing the AP Stylebook; do you see what you've made me do?—is "When in doubt, include that. Omission can hurt. Inclusion never does."