But we don't have access to the "actuality" of the world. We only have access to the world as it appears to the human mind, to the subject.
I know you're fond of Hegel's criticism of picture-thinking, isn't the above exactly this?
If we are talking about the truth of a statement or an argument, we are not talking about a picture that the individual conceives of and then attempts to match to reality. Isn't it precisely the point of Hegel's philosophy to show that such a primitive account of thought does not capture the true richness of logical thought?
See in particular:
http://www.marxists.org/reference/archive/hegel/works/sl/sl_ii.htm
The flaw in your argument is that you assume thought to be immediate, private and subjective - like a picture in the mind. But the very framework in which thought is realised and expressed is historically mediated. That is your language, your traditions, your education and so on are implicit in your thought. That is your thought is a product not merely of you the subject, but you the social, historical object. We really do have access to 'actuality'. That your form of expression is relative to your culture does not mean that the content of your expression is relative to your culture.
As Marx said, "The main defect of all hitherto-existing materialism — that of Feuerbach included — is that the Object, actuality, sensuousness, are conceived only in the form of the object, or of contemplation, but not as human sensuous activity, practice, not subjectively. Hence it happened that the active side, in opposition to materialism, was developed by idealism — but only abstractly, since, of course, idealism does not know real, sensuous activity as such. Feuerbach wants sensuous objects, differentiated from thought-objects, but he does not conceive human activity itself as objective activity. In The Essence of Christianity, he therefore regards the theoretical attitude as the only genuinely human attitude, while practice is conceived and defined only in its dirty-Jewish form of appearance. Hence he does not grasp the significance of ‘revolutionary’, of ‘practical-critical’, activity."
Note that Marx's criticism of Feuerbach's materialism consists exactly in doing what you say should not be done. That is "objectifying the subject". It was Feuerbach the contemplative materialist who failed to see the subject as object and Marx the consistent materialist who emphasised the importance of doing exactly this (as did Hegel the consistent idealist).
Note also that Marx highlights this curious connection with anti-semitism and Feuerbach's understanding of thought as mere subjectivity.