Cardboard doesn't have a terrible U-value, roughly the same as rockwool.
But that's in cellulose fibre form, rather than sheets. I doubt you could
get a tight seal with it in box form as there would be too many air gaps.
I suppose it would be possible to cut pieces to lay it flat.
I'd however worry that it would retain moisture, not what you want next to
roofing timbers. Wet insulation isn't good insulation. Commercial
insulation is also treated against fire, insects, etc, which random
cardboard won't be:
https://criticalconcrete.com/out-of-the-box-insulation-2/
If you're going to use rockwool, you could lay rockwool over the top of your
existing insulation, making a decent blanket layer. Then, if you're keeping
the cardboard, just have it as loose pieces resting on top. Although, if
the insulation is working correctly, it'll become cold and damp - sounds
like a recipe for mould to me.
But if cardboard is all the insulation you have... it works for the
homeless folks.
Theo