It seems to me that the difference between your view and mine is a difference of intuition about truth. So I think you might want to be cautious about throwing out intuition.This prompted me to think about the role of intuition more carefully.
I think intuition falls into two categories.
In the first category, we intuit truth and objectivity in the sense that we are aware that there appear to be distinct meanings (between true and false, or between objective and subjective) for these terms in certain language games. We may not initially understand what we are intuiting, but this forms the jumping off point for philosophical investigation. We're bootstrapped with these intuitive distinctions, and philosophy asks whether our intuition for a distinction in a particular language game has any broader applicability.
Intuition of the second kind seems to be trying to answer questions raised by the first form of intuition. I'm not fond of this second kind of intuition which is not so much about the presence of distinction but about specific facts invoking the distinction. This is the kind of intuition that is subject to optical illusions like the Ehrenstein illusion. It's not that there's no such thing as straight versus curved lines - that intuitive distinction (of the first kind) is clear enough to inspire formal investigation. However, it is clear that intuition of the second kind is inadequate to accurately categorize lines as straight or curved without formal methods.
I'll focus on one example. We know intuitively that the verb "to exist" has meaning in certain contexts. Suppose I say that a chair exists in the next room. You look in the next room and find no chair. You must say that no chair exists in the next room. That's the standard language game for the verb "to exist." The test for the non-existence of a thing is in not finding that thing. This is a distinction revealed to us by intuition. So, when I logically extend this game using a formal executable rule, we are compelled to find that "in-principle undetectability" is identical with "non-existence."
Here, intuition of the first kind has inspired us to devise a rigorous definition for existence.
Of course, this formal conclusion rules out all sorts of nonsense that people are nevertheless rather fond of. They reject my formal definition of existence because their intuition (of the second kind) tells them that, say, undetectable unicorns can exist. For these dissenters, existence in one game (e.g., chairs) is not the same thing as existence in another game (e.g., invisible unicorns). It's as if the edges of squares within concentric circles (as in the Ehrenstein illusion) are not really straight anymore. Well, this dissent, frankly, makes a mockery of meaning. I submit that no philosophical progress is possible if we're not going to take the time and effort to formally fix our terminology and meaning. We might as well stick to intuition of the second kind because our choice of formalities is subservient to it.
Now, as a moral relativist, I can't say people are objectively wrong to play such silly games with meaning (like leaving definitions floating). However, I think that if people really saw what they were doing, most would subjectively agree with my moral position that philosophy should be formal, and that there's no point in calling something philosophy if it's actually just intuition of the second kind. I don't need philosophy to tell me that I want unicorns to exist. I need philosophy to tell me when my intuition leads me astray.