Solve drainage issues - full process of amending clay soil/moss problem

I posted a while ago regarding our new garden and it's drainage issues. We were originally looking into land drains but are now considering trying to work the clay soil to improve it this way instead. It's a large space approx 50m x 20m so I understand the amount of labour intensive work this will be.

I've spent hours reading and there's so many conflicting processes so wanted to run this by you guys.

So we currently have very thick moss over the entire lawn, it's very spongy and matted. The soil is heavy clay to at least 3ft and drains sllloooooowly.

My first step would be to scarify as much as possible, add organic moss killer then scarify again? I know that with organic moss killer you don't always need to scarify after but because of the scale of the moss we want to remove as much as possible soon so we can reseed at some point soon. We hoped to start the first scarify as soon as the land is a little dryer so hopefully mid April at the latest? How long should we wait after applying the weed killer to scarify again?

We then plan to use a large auger type drill to drill holes into the soil very deeply, introduce organic matter (mushroom compost is suggested?) and grit to encourage drainage. We plan on doing this heavily in the water logged areas but also repeating across the lawn.

We then plan to dig in compost across the entire lawn area, I'm getting conflicting advice on whether to do this by hand or if using a rotovator is acceptable (baring in mind the compaction/drainage issues)

Main questions I have are:

how should the moss be disposed of? Does this need to be taken off site or can it be composted?

When we dig in the compost, how exactly does this work (prepare yourselves, this is definitely a silly, rookie question!) Do we literally just spread the compost then turn in with a shovel? What do we do to level it off? Wont the clay just be lumpy and ridiculous? Will a rake solve this? is a rotovator a better option?

Would it be beneficial to indroduce a clay breaker product at some point? If so, at what point in the process?

Similarly, liquid aeration, is this a good or bad idea? And what point would we add this into the process?

Then after we've done all of that, I assume we can sow grass seed, do we need top soil or will the compost and clay mix be sufficient, there seems to be mixed opinions on this!

I've gone back and forth on all of this a million times so I'm just trying to make a plan and stick to it so we actually get somewhere with improving the garden!