Is protein a big factor in strength gains? Is the calorie intake what really matters? Or both are equally important?
Both are important since strength is not strictly about neural adaptation or muscle growth alone. In most cases it is both and achieving both, requires both energy and protein.
I don’t understand the context of the question and feel the way you’re going to interpret any answer is going to lead to bad things.
Let’s say you need calories from all 3 macro nutrients for any semblance of health, let alone strength increases.
Depends the way you train. If you read works by people like Mike Mentzer and Ellington Darden, they often talked about common sense, well-balanced nutrition. This equated to basic diet guidelines for the general populace, which is completely contrary to most of what you will read about regarding nutrition for bodybuilders. The reason is their style of training, namely HIT where you typically train each body part with a single set to failure 2-3 times a week. This may equate to 20-30 mins training per week. Clearly pounding down thousands of kcals a day is not optimum on such a regime, unless fat gain is your goal. That’s why they recommended modest amounts of protein. And, to be fair, strength is often associated with HIT so they were on to something.