If the identity element e of a topological group G has a compact neighbourhood, then every element of G has a compact neighbourhood also -- so G is locally compact:
The topology of a topological group is determined by the collection of neighborhoods of the identity element of the group
I see that the cited proof does not require the Hausdorffness of G.
Can you please explain me why then a locally compact group is usually defined as a group whose underlying topological space is locally compact and also Hausdorff?
https://en.wikipedia.org/wiki/Locally_compact_group
Why necessarily Hausdorff? Is this a mere convention, or will the definition become inconsistent if the Hausdorffness is dropped?