I've had coughs that have lasted a month or more - including at the start of lockdown [#] (caught from my son, I think, who was coughing for 8 weeks or more). I've always assumed that this is because a virus can be kept at low levels, but can "hide" in the body not coming across antibodies that will harm it, but not spreading enough to create a heightened immune response??
Kinda like when you don't take a full course of antibiotics, the latent infection grows again to problem levels.
You indicate this is wrong thinking (not a suprise as it's just a null hypothesis from someone without biology training): so what is going on when you have a cough/blocked nose for weeks on end?
{# I'm very curious to find if the virus I had was Covid19; it sent me to bed with mild flu-like symptoms, gave a tight chest, had me occasionally [not persistently] coughing for weeks. No noticeable anosmia. Very little phlegm; no snot (nasal/sinus mucus).}