> if the virus and its DNA are undetectable then you can't spread it

The devil may be in the details. E.g. if a COVID test shows negative, it doesn't mean that you can't spread it. This is partly because different tests have different sensitivities.

> I'm pretty sure

FYI, without citations, it is hard to distinguish credible experts vs people on the internet saying "trust me bro".

Isn't half the selling point of antiretroviral therapy that you're no longer contagious?

https://i-base.info/u-equals-u/

U=U probably does not apply to all diseases for the reasons you mentioned though.