Apollo, Heraclitus, dialetheism and the perception of reality
The objection to dialetheism is emotional and metaphysical - the fear of the uncertainty that it appears to entail.
The desire for the comfort of certainty is part of the problem that dialetheism confronts. The desire to stand on solid ground is a by-product of the universal dichotomy that has provided the West with the confidence to achieve so much in science, technology and conquest - and so little in peace, harmony and tranquility. Epicurus' understanding that life was a journey aimed at eudæmonia has been lost, with the hubris of certainty replacing that ultimate goal with the inadequate, but more easily measurable - so, apparently more certain - goals of money and social position.
Actually, dialetheism doesn't remove certainty, mathematical truths that don't rely on non-contradiction are still there. All it does is reveal the shaky foundations of those possible truths, that are only hinted at by 'proofs' involving contradiction. It doesn't show that they are false, only that they're more weakly 'true'.
Apollo's Delphic maxim 'know thyself' and Heraclitus' 'on those who enter the same rivers, ever different waters flow' may seem contradictory, but they are complimentary. Pretending that it is the same river and you are the same man is a comfortable illusion, but it's no way to understanding.
Paradoxically, understanding the uncertainty, though initially terrifying, is more ultimately comforting than hiding behind the fragile curtain of certainty, that's so often shattered by events.
It's much more terrifying to see reality in occasional glimpses born by traumatic revelation of its arbitrariness, than to see its indifference directly, all the time.
Dialetheism doesn't reveal reality - it'd be daft to claim that! It does, though, when grasped, open the mind to perceive it more comfortably.