There will also also be tons and tons of asymptomatic people who are not infected at all but are getting tested. You folks are saying false negative in both of them combined is higher than symptomatic people.

Irrelevant case because asymptomatic people who have not been infected have nothing to do with 'false negative'. They will either be 'true negative' if test marks them as negative or 'false positive' if tests marks them as positive. So, the question of combining 'infected + asymptomatic' and 'not infected + asymptomatic' doesn't arise.

The point is that if you are infected, and you are asymptomatic then you are likely to have less viral load and you are likely to be detected as negative thus higher false negative. If you are not infected, then the negative will be 'true negative'.