Tonnes of speakers have wadding in, just not that many guitar cabs. Loads of Bass and PA speaker contain it. It was probably deemed unnecessary for guitar or not cost effective long ago.
I've tried this with a 4x12.
The schools of thought are that adding some material can effect the resonance of panels, reduce reflections and absorb some waves inside the cab, and some say the heat generated by the compression of gas (air in the cab) when the speaker moves inwards is also absorbed helping maintain internal temperature... not sure how much all of that REALLY works with guitar.
I think generally people seem to think you get more bass rather than less, or that it shifts the bass response maybe to a higher frequency. It's not that easy to google and get your head around and I don't really understand it. Looking into PA, Bass or HiFi speaker design is probably the way to research it.
If you want to try it, you can get rolls of stuff called Dacron off ebay for pretty cheap. I think I got 8oz or 9oz stuff when I try this. Or I got 30mm thick Dacron which looks thinner that what is in Gwems photo. I wouldn't use fibreglass and acoustic foams is expensive. The Dacron is easy to handle and cheap to get hold of. It's what you find inside pillows.
Plenum n Heather's point about isolating the cab from the floor should reduce bass response. I think its also about where you sit. My 4x12 sounds super dark depending on where im stood in relation to it. Usually stood in front of it so vertically way off axis.
