This isn't necessarily true. The big symptoms of almost any infectious disease are actually the body's response to the disease rather than being caused by the disease itself. Things like fever, cough, runny nose are usually (very very often) caused by the body to help fight the infection, rather than the infection itself causing them. This means that not having these symptoms isn't necessarily indicative of not having the disease or having less of it. Sometimes the body doesn't respond right.

If you're asymptomatic, you're not coughing droplets everywhere, which ought to make you less contagious

On the other hand, some people (e.g. myself) keep sneezing and coughing the whole winter every winter, so I definitely wouldn’t consider those symptoms without either headache, body ache, throat pain or fever...

that's also the thing I was wondering. All the people keep talking about asymptomatic spread, yet that should be relatively ineffective at spreading (unless it's super contagious and one cough can infect dozens). Somehow I think people are just not really honest with it I think...

Unfortunately that's not necessarily the case. I don't have the reference, but I read somewhere that asymptomatic patients may have even larger amounts of the virus, compared to symptomatic people.

At least asymptomatic patients wouldn’t cough around like symptomatic ones do. They’d "only" spread through talking.