Yeah, I agree. "Context omission is always subjective" seems like a wrong way to put it. Let's split facts into assertions about a model/approximation of the world being accurate (usually implicit), and an assertion about what the truth is within that model (explicit). (That split isn't always a clear bright line, btw.)

The first part is implicit, and thus 1. less visible, 2. more fluid, in that in an argument, you can often pretend you had a different context in mind later, or your intelocutor can have a very incompatible one in mind. Hence the need for trust and compromise. But it's not really any more subjective than the second part - the choice of model is very much like the choice of fact/assertion/observation within the model: it strongly depends on the world, we can tell it is intersubjective, people tend to agree on it independently, we can call it "correct" or "wrong", etc. That's all closer to what we usually call objective, like "the sky is blue", unlike "I like anchovies", even though you always need context even for objective claims ("not at night, obviously").