Rock hard lawn

A year ago part of our back yard was dug up for a retaining wall. The company laid ready lawn after work was completed. The ground under the lawn is rock hard after having heavy machinery working on it and a year on the lawn hasn't really taken. Where they laid topsoil to even out the ground and along the edge of the retaining wall the lawn looks fantastic but everywhere else is brown, even in winter and we are starting to get quite a few bald spots. You can tell the soil underneath is rock hard as the water doesn't really soak into the ground.
Airating the ground is not an option, we have tried and you need a sledgehammer to get the prongs further than the depth of the ready lawn and bend most of the prongs in the process.
Googling this problem comes up with lots of options but I would like to know if anyone has had this issue and what they did to rectify it besides digging up the lawn and starting again.
Any advice would be hugely appreciated.

Green concrete would be perfect

Lol. Mowing the lawn for 20 years will be less work than trying to get concrete up our drive. Its steep and we are a back section. When the concrete was poured for the retaining wall they needed 2 pumps to pump it in as truck cant get up drive so they had to pump it from the street AND over our 2 storey house.
We have no money for that so would have to barrow it up. Mixing it on site would be just as much work as we only have a meter wide gate so all the sand etc would also have to be barrowed in.

If it really bothers you, then you would need a hoe or similar to dig it up, add compost and re do it all.

Or just let the kikuyu grow, it handles bad soil and low water.

spread heaps of garden lime & gypsum over the lawn. it will take about one year to see the results tho

Ditto that will help in time

Barrow in enough compost and earth with high humus content to cover the area to 100mm depth. Then get in a stump grinding machine and get it to dig over the whole area and, in the process, mixing the compacted earth up with the barrowed in compost and earth. Level, compact and sow new lawn, covering with light cloth eg. frost cloth. Water daily late evening and early morning until new grass is 75 mm high then regular light trims until the following spring. Best lawns result from an Autumn laid lawn. The stump grinder will not even notice that it is working on very hard compacted ground and the earth will not cause any wear on the blade so it is only a small cost to run the machine and an hourly rate for labour. Better than any rotary hoe.

Thanks guys. I will go and get some tomorrow. Don't mind slow results just happy to know our lawn can be saved