Exactly. The problem with self testing is trust.
If you get tested by an authority - a doctor - then there is only so much discussion to be had. Whereas if you are asked to test yourself in the morning, who's going to be able to verify your claimed outcome?
Consider STD's and - in particular - HIV. Even with medical testing, the first basic lesson any teenager (should) get(s) is to use a condom if you don't know your partner.
Moreover, it's a good thing that many legal systems (a) only grant doctors the ability to issue sick leave and (b) protect the rights of those who are issued sick leave. Labor laws being byzantine as they are, allowing employers to test their workers at their own discretion poses a wide range of prickly legal issues.
The fact that South Korea or China (arguably) succeeded in curbing their epidemic happened because their public governance puts the collective above personal rights and people are totally fine with that because that's a deep part of the moral standard of their culture.
Current mass quarantine exercises are painful in the short term, but if you can keep them up for significant time, it might be possible to safeguard liberties and protections that aren't self-evident. You could curtail those exercises and favor self testing in order to open up the economy quicker, but doing that in a meaningful way requires that we all are prepared to compromise on some of those protections and rights, with all the unintended societal consequences that might bring.