We are in a newish house,  and looking to improve our yard in both getting rid of standing water and improving our soil to get grass and other plants to grow. I am curious on any and all advice!

The problems:
I believe the water issues are twofold. Of poor grading under the deck stairs, the bottom of the stairs are definitely a low spot, and poor soil drainage. The photos show the pond in the low spot and the water standing in most of the yard near the house. 
A bit of history on the house, they had to over excavate our basement due to expansive soil, I believe said soil was spread out and used to grade around the house. Our landscaper brought in what they said was 2-3" of black dirt (it did not end up being that much everywhere) so that grass could be seeded.
I had asked about tilling dirt or compost into the clay heavy soil instead of layering black dirt on top but the landscaper told us they did not think that necessary. 
I am not 100% thrilled with the situation but it is our starting point nonetheless.

My initial thoughts: 
At any rate at this point I'm thinking a drain in the low spot by the stairs, maybe a loop of French drain around the stairs? Running it with tile to a nearby ditch and towards the pond is pretty doable. 
I don't think any of our downspouts are affecting this area specifically but I plan to trench those remaining into the same ditch. 

Also aerate the crap out of the yard and potentially top dress with compost or black dirt to attempt to improve the soil drainage? And hopefully let grass (and eventually other plants) grow better.

My questions:
I know I don't have any yard slopes etc, but I am curious if the French drain (or other drain designs, I am open to recommendations) will help drain the low spot quickly enough. Or if regrading is needed. I'd like to avoid redoing the stairs, but if we need to raise the ground under those it is not technically impossible . 
We are also thinking of expanding the patio out eventually, so I figure I can use this planned tile for a drain integrated with the stones at that point. 

As far as the soil, I have not done any tests, but I think we are having issues with the layers of materials, whatever the original was, the clay heavy dug up under the house, then black dirt on top. My gut tells me the grass roots are not penetrating the clay layer, and would benefit from tilling or discing the layers together. At that point I think soil amendment is in the question at the same time, so whether that be compost, dirt, sand or something else I would need to research as well. 


I appreciate any and all recommendations, we have access to a small tractor, bucket and box blade. a mini excavator, and a side by side,  I am thinking we could use at least a core aerator to get some drainage improvement, but I do not know how much further we need to take it. 

Thank you all! 

My puddle of a backyard