A quick look at the meaning of "ausschlafen" and its brothers "verschlafen" and "einschlafen". Also: some grammar... yawn!
ausschlafen, verschlafen, einschlafen, schlapp, schlaff, der Schlaf, der Lappen
The core idea of this root was:
weak, lack of tension, hanging down, gliding down
According to DWDS.de, the main etymological source I am using, it’s the root of sleep and the Latin labi, which was about weak, collapsing.
(English sources do not agree with this connection and list the root of sleep as sleb-, but the core idea of weak is the same)
The Latin labi is the origin of words like lapse, collapse, labor and elaborate.
Other members are sleep and lip.