Too many things can go wrong. Leave it to the professionals.
I am a huge advocate for having more test kits available. We need tens of millions of people being tested by next September so we know what is going on with the disease. Some have considered home test kits, see No COVID-19 testing at home yet but quicker options coming.
Test kits remain an issue in this nation, but untrained people at home doing tests is not a good idea -- in my "I'm not a doctor" opinion.
The lack of test kits started the mess we are in and it continues to dominate why we are in the position we are in -- plus, the slow reaction to what early detection there was and watching what was going on in China.