People already get further testing. People already get second opinions. In the time you are changing your lifestyle (a good option for helping treatment, I agree) and increasing monitoring you could already be progressing beyond treatability.
You can argue about whether it's alright to tell sub-1.5 million people each year who don't have cancer that they have cancer (I don't think it is, but whatever). But again you can't disregard the non-insignificant number of people who get false positives.
edit: Actually, not "whatever". You probably shouldn't tell sub-1.5 million people each year who don't have cancer that they have cancer.
It just doesn't need to be as dire as what you are projecting.
That's actually a very good argument for early testing.
And needlessly scaring a some people into eating better and exercising more is about as bad as not screening them at all to avoid needless anxiety.
Heart disease is a huge deal. It ruins your life. (Maybe not in all caps.) Nevertheless, we regularly test for it, including with techniques with high rates of false positives. Few panic because almost everyone knows someone who got a false positive and didn't promptly keel over and die.
If healthy adults are screened for cancer, there will be cultural memory of people who got a false positive. The response to a positive test result won't be chemo. It will be further testing, if non-invasive, or suggestions of lifestyle changes and increased monitoring of the suspected system.
People aren't too stupid to understand an early test with a high false positive rate. People with family histories of cancer don't wake up every morning screaming. Some may. But barring everyone because some people will overreact is why orthodox health policy is losing public trust.