Can you use push or pop traffic for search arbitrage?
Generally not with Google-based feeds, and often not with others. Push notifications and pop-unders are cheap, but they produce many accidental and low-intent visits and carry a high share of invalid traffic. Feed providers publish lists of approved sources, and sending unapproved traffic is one of the quickest ways to lose a feed and unpaid earnings.
Pop traffic opens a page the person did not ask for. Push notification traffic arrives as a browser notification, often from a subscription the person does not remember accepting. Neither starts with a deliberate choice, which is the thing search advertisers pay for. Networks selling this inventory also have more bot traffic and domain redirect traffic mixed in.
The temptation is to blend a little cheap traffic into a clean campaign to lift margin. That is traffic blending, and it is exactly what quality systems look for. If a provider does allow a particular source on a non-Google feed, get that approval in writing, isolate it on its own Channel ID, and watch finalised revenue closely before spending more.