I think it could be argued that a pursuit of objectivity is based on valuing objectivity; and there is no way to answer why we should value it. We can say things like "well, if you believe objectively true things, you're more likely to stay alive because you won't do irrational nonsense things like leaping in front of cars, believing it wouldn't harm you but instead make you stop being hungry." Or whatever.
But then there will always be a microcosm: say you're hungry. Well why value not being hungry?
"Because then you won't die."
Why value not dying?
It sounds silly, but we never reach an end here. At the end of the day we have to just accept that we happen to value particular things for whatever reason, and we didn't reason our way into those values.
What does it mean to value something? First and foremost, value is dependent on, and specific to, an individual sentient being. Values are an expression of a specific being's complex cognitive state (or states, however you want to frame it). Reality exists independent from what any one sentient being thinks about it, and certainly independent from whether or not that particular sentient being exists. All this outside of what the sentient being thinks would be the “objectively true things” to which you refer.
I would argue that to ask if something is objectively true would not be asking a subjective question, but would be asking an objective question. Is this thing or condition a true and existent thing, regardless of what I feel and regardless of whether or not I am here to observe or think about it.
To your comment that there is no way to answer why we should value objectivity, or objective knowledge, you contradicted yourself by providing an example of why one might value objectivity - to not get hit by cars. And it is not infinitely recursive. We are pre-wired with an instinct to survive, to continue to exist.
As to your position that we don't reason our way into values, I would argue that how we acquire values is a multifactorial process. We are born with a set of pre-wired behavioral instincts that can be said to provide us with immediate, reflexive value responses. In addition, as we grow and develop, we add to our preset instructions the knowledge that we gain through our perception and experience. Our experience allows us to develop reasoned expectations about the world around us, and it is this reasoned understanding that informs our values, judgements, decisions. On top of these two influences on values, we have socialization and indoctrination provided by the environment and society in which we are born and live.
That we are aware of an external objective reality is a product of reasoned expectation based on experience, it is not something chosen or pursued. Whether we choose to ignore or deny part or all of that objective reality would be a subjective value choice.
