No, that is not generally the case. You have flare only in certain conditions, but not always. In diffuse daylight or with the light source behind you you won't have any flare. I have used enough old non-coated and single coated lenses to know their characteristics.
See explanation above. And please don't ignore the physical fact that lens coating is increasing and improving light transmission through the lens.
Your statement that uncoated lenses will give you "additional light" is not true.
No, your problem is that you don't understand that uncoated lenses cannot solve the original problem with this film. And that you unfortunately don't know what a characteristic curve really is and what it is showing. And your object contrast / subject contrast is determined by the lighting conditions on that object, and not by the lens.
If you could significantly influence object contrast by lens choice our photography techniques would be very, very different to what we are using for decades!
Glass does reflect light, if you`re outside looking at a window pane you will find the environment to be visible in the window - though windows are intended to let light into the house and not reflect it back. But as you also don`t want everybody to be able to look inside your house, windows reflecting some light are not a real problem.
But with lens elements it is problematic, if there are too strong reflections. That's why they use optical glass which does reflect fewer light than window glass - while window glass does reflect about 20-30% of incomming light, optical glass does reflect only about 6-12% - let`s take the average value and say its about 9%.
9% is what each element of a uncoated lens does reflect, with single-coating did reduce reflections to about 3% and multi-coated is about 1% per lens element.
This means as soon as there is light passing through the lens, there will be reflections, no matter what light or light source there is - even with a multicoated lens. If a light source does shine directly into the lens, there will be visible artifacts which usually are called lens flare, but there also is a different kind of flare which could be described as an even fog laying above the entire picture - and because of that isn`t that obvious or not obvious at all.
This is what i mean when i was talking about flare, not obvious reflections of the aperture or multiple images of light sources. Think of it as a fogged lens, it will produce flare even if there is diffuse light comming from behind.
Now as a single coated or uncoated lens does produce flare (fog) any time, there will be additional light be shed on the shadows during exposure. Light that would not be shed (respectively much less) onto the shadows with a multicoated lens, as this does produce way fewer flare (fog).
This additional light on the shadows of course is not comming out of nowhere. A single- or uncoated lens does not have higher light-transmission and there are no extra openings in the lens barrel to let in additional light, no this flare (fog) is created by the lens elements reflecting light. That's what i meant by "stealing light from the bright areas of the subject". Light of the bright areas will be reflected between the single elements, it will be converted to flare (fog) and spread even above the entire neg.
If we now only take a triplet, a single-coated lens will produce 3x3%=9% flare (fog) of the light passing the lens, while an uncoated triplet will produce 3x9%=27% flare (fog) of the light passing the lens.
That`s quite some flare and this should be able to brighten up shadows on the neg inside the camera.
And if shadows are brightened up, there is fewer contrast on the neg. Therefore a lens is able to influence subject contrast, any lens does reduce subject contrast as there is no lossless lens - the question is how much the lens does reduce contrast. And a multicoated lens does reduce contrast a lot less than an uncoated lens.
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So i`m comming back to my theory: If a film like P30 does drop shadows, what would happen if you did not use a multicoated lens, but a lens producing more flare (fog) which does brighten up shadows?