Few things annoy me more than when "safety" is used by some neanderthal to bully or abuse those he sees as 'under' him. Nor have I much use for when 'safety' is used as a BS excuse to cover for a discretionary management decision. If a firm wants to have a dress code, come out and say it, and provide a suitable alternative ... don't blame "OSHA" just because you don't like something.
Just as bad are bureaucratic responses such as the one (nice link, BTW) provided by OSHA when asked. The clown freely admits that there's no prohibition - but then goes on to say 'we prefer ...' Hell, I 'prefer' winning lottery tickets - but for some reason they don't let me return the losers. Imagine that. That's what 'preferences' mean.
Let's cut to the chase, and look at the ANSI test standards. An impact is provided to a single point dead center at the top of the hard hat. That's it. There's NO assessment of the ability of the hat to remain on your head, block rain, or take hits anywhere else. All that's expected is that the suspension distribute the force of the impact all over the test 'head.' The different ANSI helmet standards (for bicycle helmets, motorcycle helmets, etc.) all differ only in the shapes of the impactor and the test head. (Which is why it's hard to say if a skateboard helmet would qualify as a hard hat).
With that simple test in mind, it's pretty clear that any hat or hood will not affect the test - unless it has some feature (like the button on the top of a baseball cap) that would concentrate the forces at one place. FWIW, none of the liners marketed for use with hard hats have ever been tested with any hard hats. We 'know' their effect no more than we 'know' the effect any other hood might have on the performance of the hard hat.
That's not to give a blanket endorsement to hoods. Here I expect some intellegence to apply. It is conceivable that a hood might - I emphasize might - present other hazards. For example, by getting caught in machinery or by obstructing side vision. It would be an exceptional circumstance were this to happen.
Finally, let's look at someone with extensive experience wearing helmets: the military. They have all manner of headwear - including, now, a Sikh with a turban - that they've managed to wear under helmets without problems. They.ve managed to get hoods on their jackets that do not block side vision, and that stow out of the way.