Yes, they do... as long as they find enough food and water. Budgies are much more adaptable and smarter than people give them credit for. Couldn't you contact somebody nearby to either try to trap it or, at least, leave food and water out for him? Maybe somebody in a nearby rescue? It doesn't have to be a bird rescue, dog and cat rescue volunteers usually love all animals and will try to help if they can.
Now, more lost and/or found birds.
Tiel was lost but found woman that had hair similar to her own owner (clever birdy!)
https://wtop.com/dc/2020/04/dc-woman-fi ... -find-her/