Why doesn't the night sky keep on getting brighter and brighter every day? I think this should happen because of the below logic.

Consider Day 0 of planet earth. There is no light coming from any surrounding star that has reached our planet and hence its pitch dark.
On Day 1, the light of 1 star reaches us. And once it reaches us it will keep on coming continuously as it's not a pulse, it's just a continuous stream. Let us think that it's a 'string' or 'thread' say which starts from that star and reaches our planet and so we now have this connection.
On Day 2, light from another star reaches us. Again there is 1 more 'thread' which has now connected that star with our planet.
On subsequent days (and in between) more and more such threads get connected with our planet. And unless earlier stars that made this connection with our planet dies, the connections will only grow in number.
And so I would think that the night sky should become brighter and brighter every day.

Why doesn't this happen?
This assumes that the stars formed at the same time as the Earth.  This is not the case.  Stars had been forming for billions of years since the Earth formed.
The Earth didn't form in a single day, but for convenience let's pick one Early day in the Earth's formation as &quot;day 0&quot;.
On day 0 light from many stars that had been travelling for years reaches Earth. The night sky is full of stars.
On day 1 light from those same stars that was emitted one day later reaches Earth, the sky brightness doesn't change.
On day 2 the same and so on.
Over time some stars will have moved nearer, some will have moved away, some stars will have reached the end of their life and some will have been born. But, on average there is no change.