I'm not arguing for relativism. Yelling at someone to get out of the way of a moving car would be a command rather than a statement that's either true or false. The command might imply the statement that there's a car coming now, but that statement would amount to little more than tracking. The label "car" adds to our understanding when it's associated with a conception or a model, and that would be the subjective, human element.
It's not so much what we call "objective truth" that I'm rejecting, but the "conceit" of pure objectivity that I think should go. The question is how to make sense of what we call objective truth. Is it purely objective or always partly subjective? The more meaningful the symbols we use to present the truth, and the more they add to our understanding, the more subjective the statement will be.
I'm defending a pragmatic theory of objective truth. So objective truth will be that which is useful in different ways than subjective truth.
Human conceptions may be supreme according to what we know is actually the case, but in trying to understand what we're doing at the philosophical level, we're free to contemplate possibilities, probabilities, and hypothetical scenarios, as in thought experiments. Is it reasonable to think we couldn't possibly evolve into a superior, transhuman species which would deem its former, mere human conceptions as being as narrowminded as the ones we presently associate, say, with squirrel thoughts about cars?