Wet spots on covered patio after rain

Yes, this should not be a problem. Under your patio should be various layers of gravel that should drain the water well. If the rain unsettles anything, you can always tamp it out while doing your repairs, but I would not expect much damage.

Concrete is pretty hygroscopic, meaning it attracts water - to a degree in my experience. Capillary action is the classic way concrete attracts water, but if it is cooler than the air it can and will condense moisture from the air. And concrete holds water in and under pretty well: if you were to dig under it, probably not dry.

If accumulation only happens in some spots, are they depressions, or lower than other adjacent areas? That would be why it accumulates there. Another thing to check is whether those specific spots are cooler than other areas so you have local action going on (sounds crazy, but you've exhausted a lot of other ideas, now it's time for outta the box thinking.) Use a pretty sensitive digital thermometer (measuring portions of a degree) to check.

Which brings up another point: is the air in your basement somehow a bit moist? Have you checked the humidity level, esp. in various places (dry here, moister there). Is there any possible air-current that could move air over specific spots, or around them so they don't dry?