One reason we believe this is that we can typically detect virus quantities way too small to reliably replicate in culture, and believe that in vivo infection is even harder than in culture (because of innate immune response and the body not being composed just of the most susceptible cells).

You generally can't measure infectious dose directly without a highly unethical challenge study. You can sometimes know concentrations that did or didn't result in infection in various real world scenarios, and sometimes you have circumstantial evidence (e.g. you can know how much virus a person sheds, and what proportion of a room with certain ventilation quantities got infected).

The usual way of achieving undetectable viral load is via anti-retroviral drugs and I think this works in most people, at least so long as they get early enough treatment and keep getting it. The main reason to measure viral load in the first place is to get an idea of how well the drugs are working and if they need to be changed.

People on ART also often (perhaps usually) have undetectable viral loads