No. If someone wakes up and feels they've got something worse than the normal cold, they phone up and get someone to come out and test them (and the rest of the household). They all wait for the test results to come back. If those test results indicate something that poses a threat (or something new or unknown) then they isolate. If its not something that poses a threat, then the decision is up to them (so if its just normal flu for instance, they find out its just flu and can then decide what they do with that; if its nothing, they are told its nothing).
This is people taking one or two sick days, when they are sick, and reporting they are sick to the authorities. Their employer should (and would be told to) cover that as they should do any sick day, but for where this isn't possible (self-employed people for example, or gig workers), then yes they should be helped to do that - given that its in the interest of everyone in society that we are able to pick these things up early enough.
If people take the piss with the system then that will show up too - you'll have the same people and/or the same addresses coming up repeatedly. If people lie about having done it, then I'd hope the employer would have some way of confirming whether or not a test has taken place (even if its just ringing up and checking a reference number).
As for whether I think that has any sense of realism - yes, I do. Most people will call 999 if they need help; they'll call 111 or go to the GP / A&E if they need medical help. If that help results in advice to do something, most people will try to do it if they can. People already take sick days from work. We already do most of this, in short - we just need to establish a test, track, trace and isolate system that can work and which people will use.