-
https://www.lawnsmith.co.uk/lawn-care-videos/aerate-rake-scarify/spike-core-aerate.htm
No need to dig, get one of these and a bag of sand. Grass grows ok on clay, just keep mowing.
-
You would need to line your augered holes with a geotextile to prevent the gaps between your grit particles filling up with fine soil particles.
Are you 'West'?
I have a 5-inch auger you can borrow.
(Augering stone-free London Clay is a weirdly satisfying activity.
You bring up inch thick 'pancakes' of yellow/orange clay).
My lawn is mostly pretty damp clay London stuff. I've read that you can try to improve drainage by digging grit and organics into it. I can't do that (as it's my lawn, and it's covered in grass and moss...).
I thought about getting one of those big augers that you use for 'drilling' fence post holes, and filling the resultant holes up with grit (topped off with soil / grass seed) to get a few big drainage holes that would hopefully reach through the clay into better draining stuff. Does this sound plausible / useful?