There are destinations where United does not sell basic economy fares, such as Dubai and South Africa. For some Asian destinations (Japan), they only sell economy nonchangeable fares, which are essentially BE prices but without the seat assignment/upgrade restrictions. If you're looking for a miles upgrade, that's where you should look. Granted, mileage upgrades also will come with hefty co-pays, which it sounds like you might be averse to paying.

Also, depending on whether you have status and if so what level of status, it may be hard to clear an upgrade into Polaris, although there are certainly routes with lighter loads where you might stand a better chance.