Consider what each skill actually demands of you, cognitively.
Reading and listening are receptive skills. Their job is recognition: a word or structure arrives from outside, and you match it against something already stored. Recognition is genuinely valuable — and genuinely forgiving. Context fills gaps. You can understand a sentence while being unable to produce a single clause of it. You can follow a film with a vocabulary you could never deploy. Receptive skill is real skill, but it lets you coast, and it will let you coast indefinitely.
Writing is productive, which is a different demand entirely — now you must retrieve, assemble, and construct. But writing is production with the clock stopped. You can stare at the sentence. You can look a word up. You can revise. Writing trains real production machinery, slowly, in laboratory conditions.
Speaking is production with the clock running. To speak, you must retrieve vocabulary in milliseconds, assemble grammar in real time, monitor your own output, and — because speaking almost never happens alone — process incoming speech in the gaps between your own turns. Speaking is the only one of the four skills that recruits the other three simultaneously and under pressure. When you speak, you are retrieving what reading stored, constructing what writing rehearsed, and parsing what listening trained, all at once, with a human being waiting.
This is the finding at the heart of what researchers call the output hypothesis: producing language does something that no amount of comprehending it can do. Comprehension lets you skim meaning off the top of a sentence without processing its machinery. Production forces you through the machinery — there is no other way out of your mouth. Output isn't just evidence of learning. Output is where a particular kind of learning happens, and it happens nowhere else.
So the four corners do reinforce each other — but downhill. Speaking pours strength into all three of the others, because it exercises all three of the others. The reverse trickle exists, but it is a trickle. Years of reading raise your speaking only minutely, which is not a theory but an observation available to millions of learners who have run the experiment on themselves.