Grey clay is usually alkaline [asris.csiro.au] If i was dealing with it i would adjust the ph to around 6.5 with elemental sulfur, adopt a lasanga style garden and build compost in place with only tilling in at the end of the season each year.

It depends on the nature of the clay as well as the alkalinity of the soil. With acid soils you are better off using lime or egg shells to moderate it. Wood ash does wonders to but with the charcoal not without. You would be best to use coarse sand or pea gravel as well. For these soil choose compost over peat or manure.
For alkaline soils mulch, peat and some manure works well. Be aware that cattle manure is often far too, rich for most soil, horse however works quite well; just prepare to deal with the weeds it brings.

Peat, sand and humus will loosen up that clay

I'm not an expert, but I would imagine sand might help.

Or this crushed up shell stuff.. Vermiculite? Something like that.

Cottonseed hull mulch helps a lot when mixed in thoroughly. Also diatomaceous earth helps.

I just go raised beds

Rototill sand and compost into the clay then leave a drainage point from the garden via a french drain if you have some slope to the property.

We have sandy soil here in Florida. But I know you have to get some organic mater in there. Probably won’t hurt to mix in some sand rock dust and bio char.