In principle, I find partially integrated garages in the house much more harmonious in terms of appearance, but as already mentioned, you bring a dirty area into the house.
Regarding this property, the southwest corner of the house would be far too precious for a garage for me. Here you have the evening sun in summer, and the terrace is already shaded.
Kaho’s idea of using the basement for living spaces and maybe having a connection (stairs in the garden) between the two terraces is charming.
If the driveway must absolutely be from the west, you could also consider a protruding garage with a roof terrace accessible from the ground floor. Both concepts would allow placing the stairs one above the other in a sensible position within the house.