Grass disappearing and moss taking over

[GettingOnWithItAll]

I have a south-facing garden with a 1 in 10 slope with a flat concrete path and a Bay tree at the bottom. Below the first ft/0.3m the soil is very sandy. The soil at the bottom is wet when it rains, but there’s no flooding. South of London, no strange ground features.

Three years ago I took up the old turf, added some spare more-claylike soil, evened out and reseeded. I got the usual odd weed and bit of moss on the Eastern edge for a couple of years but this winter I’ve had terrible problems with:
a. The grass disappearing on a large swathe of the bottom and being replaced with moss, particularly under the bit in the shade of the Bay tree, which is also used by the local cats, foxes and badgers that wander in from the wooded railway embankment behind.
B. Patches of yellow grass in the shape of boots coming from the house patio/steps replaced by limestone slabs shortly before redoing the lawn.

Yes, I can put down moss killer now and then in a month’s time add soil improvers and reseed the grass. However, I’m being pushed into a quick decision as I have a day to decide if I want free hard core for alternatives (I have spare cement and ballast if I want to give up and concrete for eg a kids play area).

I’d also like to knock down the Bay tree but my wife won’t have it.

Can someone please advise if I’ll be wasting my time trying for a good lawn and/or anything special I can try?

The images show the garden after I did a bit of scarifying on some moss.

 

[Wayners]

Moss killer. Scarifier or Rake. Seed. Lots of firtliser through the summer with water and cut the grass on long setting keeping the moss in the dark so will die. That's it.

People don't feed or water and cut too short

 

[elisa123]

As above and you could lower the height of the bay tree to let a bit more light onto the grass.

 

[Mike13]

The Bay tree would survive a crew cut, and would crop well for adding dried leaves for cooking.