So it seems we agree on the meaning of the word.
And certainly saying "apply leverage" would be fine. But I can't see the -age ending as a problem. What about haemorage? That's a noun and a verb, surely (even if I've misspelled it, as I'm sure I have). We could say "suffer/have a haemorage" with the same meaninng, but it's simpler to use a one-word verb.
That said, I can't imagine using the verb "leverage" myself, any more than I'd say "to advantage" - I'd probably use a completely different phrase - but I don't object when others do.