The limbic system controls a huge swath of things throughout the body, but the damage seems concentrated in areas like sensation, memory, and emotional control. Which makes sense, since loss of smell, brain fog and memory loss, and anxiety seem to be some of the more common long covid symptoms.
The extent varies but I don't think any system is badly damaged enough to bring it down entirely.
And yes, it has to physically enter the brain to do its damage (cytokine storm notwithstanding). Covid is not the only virus able to cross the blood-brain-barrier and take up shop (hello herpes, you mischievous little mink) and several of them can cause damage. For viruses like covid, entry is normally through the olfactory area, as the holes in the cribiform plate and the noses direct nerve line to the hippocampus make it a weakness. If it breaches there, not even the most determined astrocyte or hungry macrophage can stop it.
What makes covid sort of interesting is the extent and spread of the damage. It hits a lot of places. This is because it has a fun little spike that just so happens to match up with one of the most common receptor sites in the body, which occurs frequently in neuromuscular junctions and, you guessed it, the limbic system.
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