I've seen this play out with people I know, and it's frustrating because they often feel dismissed - like their doctor doesn't believe them or thinks they're just deconditioned. The thing is, your brain fog and fatigue at this point aren't just "you're out of shape." Something's actually different in how your body is functioning months out from infection.
The answers above touch on inflammation and immune dysregulation, which are both real parts of it, but the picture is messier than any single mechanism. What researchers are finding is that long COVID probably involves multiple things happening at once - and different people might have different dominant problems. Some folks do have ongoing inflammation markers in their bloodwork. Others show signs that their mitochondria (the power plants of your cells) aren't working as efficiently, which would absolutely explain persistent fatigue. There's also evidence that for some people, the virus damages the lining of blood vessels or leaves microclots, reducing oxygen flow. Your brain needs a lot of oxygen, so that could be your brain fog right there.
The hard part is that we still don't have one test that says "yes, you have long COVID." It's diagnosed by ruling things out and seeing if your symptoms fit the pattern. What I'd push for with your doctor is getting checked for things like B12 and iron levels, thyroid function, and maybe asking about a referral to someone who specifically sees long COVID patients - some clinics have popped up in various places that track this stuff more closely. In the meantime, pushing yourself hard usually makes it worse, not better, so pacing matters more than "just powering through.