I hear you. When you're trying to write one big test that verifies the whole code flow or whatever, it can be HELL, especially if the code has been written in a way that makes it difficult to write a robust test.
God, big mocks are the WORST. It might not be applicable in your case, but I far prefer doing some setup and teardown so that I'm actually making the network request, against some test endpoint that I set up in the setup stage. That way you know the issues aren't cropping up due to some mocking nonsense going wrong.
Asserting that some arbitrary numbers match can be quite fragile, as I'm sure you've experienced. But if the code itsself had been written in such a way that you had an easier assertion to make, well, winner!
Its all easier said than done, of course, but your colleagues having given up on testing because they're bad at it is kinda disheartening I bet. How are you gonna get good at it if you don't do it! :D