That is, if the closest one we can observe is 1600 light years away, are we not observing these things as they were 1600 years ago? How do we know black holes didn’t just pop up for a brief time window, and now they are gone?
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The city I live in is pretty old. Due to its age, some buildings in the historical centre have slightly slanted floors. I’ve been in a house with such a floor. It’s not so that you’ll immediately notice, but put a tennis ball on the floor and it’ll roll away from you.
In other words, look at the floor and you won’t say something’s off about it. Look at the ball and you know not only will you find out that it is slanted, you will also quickly learn which side of the room is higher than its opposite side.
Same with black holes. Even if we can’t see them, we can see the effect they have on the things around them.
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