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Hang in there. If I'm reading between the lines correctly, I take it you're frustrated because you weren't retested because of your negative result, while the wife and kid were positive. Assumed you were all tested using an in-home Antigen test, correct? Did you retest yourself or try to get retested? If additional testing was unavailable for some reason, yeah, the standard "Quarantine" rules would apply and that would be that.
One of the big holes in the home testing strategy is actual result reporting. Very few, if any, report results, be they negative or positive. FDA is trying to link home tests with using a phone APP to get results so the data goes automatically to state officials, but I think they're taking a leak on the windward side of the boat. I've run over a dozen tests at home and have never reported any of them to the State CDC. FYI, I just ordered 3 home antigen test kits from Sam's Club with home delivery. Not sure if you still can get them there if anyone is interested.
As a matter of protocol, nowhere near every positive result goes on for typing. It's complex testing that can only be run in a few special labs. Each state takes a representative sampling and sends them out, that's why you see the reported numbers as percentages. I'm not even sure if your sample was sent out for variant testing you would be able to find out with the result was. That adds a level of complexity from the sample chain of custody that is not necessary.
Now IF there ever are different treatment regimes for the different variants, then it would be critical info, but since that is not currently the case there's no need right now.