For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in thinking we can prove this by seeing a topological group as an algebra over the Ass-operad and applying homology everywhere?
You can do that. If an operad O acts on a space X, then the structure maps
O(n) x Xn -> X
induce homology operations
In particular if O is the associative operad (so O(n) are discrete) then this structure reduces to the Pontrjagin ring structure.