Ugh. The more I think about it, the more it becomes a problem of resolution and exposure time.
Resolution, because you really want to zoom in and know if you're looking at 2 things close together, or 1 thing.
Exposure time because the things you're looking at are up to 13.8 billion light years away and not too many of their photons happened to be pointed directly where you were eventually going to put your eye. You need to look for a long, long time to get an image, and you need to have a sophisticated understanding of "stillness" during that exposure.
(2,000 exposures / 500 hours of exposure time = 4 hours per exposure on average)

It's complicated by the fact that those deep field surveys are intentionally looking at "black" areas of the night sky. Whatever the percentage of those surveys, that's adding to the percentage of the night sky that is already blocked by something closer.

So.

What %-age of the night sky is visible stars and galaxies?
Should we really count those?
Is it in the nature of your question to assume we're not trapped on a planet in some random galaxy, but rather somehow observing from a (much more common, statistically speaking) empty region of intergalactic space?