Except the body design most certainly does not resemble a lion’s in the slightest. So while putting a lion head on a body (as in the illustration) may satisfy certain assumptions, it does not change the plain facts, to wit the body is not that of a lion.
As for the name - ‘sphinx’ is not the name the Egyptians used for that object. I’d have to wade through a thick book to find the actual name, and it is too early by half in the morning. However, the hieroglyph referred to Anubis; sorry I cannot recall it off hand at the moment - perhaps later after I make some coffee.
The general body shape is not eroded, just the details show degradation.
What we see of the body is largely made up of laid blocks, some of which go back to pharaonic times; the paws in particular are not carved out of the bedrock. The name "sphinx derives from a distorted Greek translation of the Egyptian word shesep-anhk, 'living image'..." [Miroslav Verner, "The Pyramids", p 237)
After Thutmose IV cleared away the sand during the 18th dynasty he memorialized his own effort with the Dream Stele. [snip] From that point on the cult of the Sphinx -- which was called Haremakhet, "Horus in the horizon" (Greek Harmachis) -- rapidly grew in significance. [/snip, ibid]