So am I correct in saying that, in Esperanto, gerunds are treated as infinitives?
Not always. You can also change a verb to a noun by replacing -i with -ado (kuirado gravas = cooking is important). But whenever you have a multi-verb construction like "finishes eating", you're usually going to use the infinite for the second verb.
If you say "My wife finishes to eat the bread" you see the infinitive too :)