Additive noise meaning
noises gets added therefore it is additive

correct!

which make me think that noises are not destructive

incorrect :(
simple thought experiment: You flip a fair coin $X$ (Head = -1 / Tail = 1) and tell me the result. The entropy here is 1 bit, i.e. the (expected) information ($I(X=\xi) = -\log_2 \left[P(X=\xi)\right]$) of each outcome is 1 bit.
Then there's additive noise $N$ that takes one of the values $\{-2,0,+2\}$ with equal probability.
When you receive a -1, you can't know whether the coin was Head and there was 0 noise, or the coin was Tail and there was -2 noise. Both are equally likely!¹
So, your additive noise is absolutely able to destroy information and hence is very destructive to your signal.
If you're more coming from a wireless communications background: Your $X\in\{-1,+1\}$ can be interpreted as BPSK. Now you see how even benign Gaussian noise destroys your reception when its sign is the opposite of your transmit symbol!

¹ we can even formalize that. Since $X$ (2 options) and $N$ (3 options) are independent, and each of them equidistributed, there's six possible combinations, each of them equally likely
 X | N  | Y = X+N
------------------
-1 | -2 | -3
-1 |  0 | -1
-1 | +2 | +1
+1 | -2 | -1 
+1 |  0 | +1
+1 | +2 | +3

Thus, we have four possible outcomes for the sum of signal and additive noise, -3, -1, +1 and +3.

If we see +3 or -3, we get 1 bit of the 1 bit info in the coin toss. (It must have been +1, otherwise we couldn't get +3, or -1 for -3, respectively.) That happens in 2 out of 6 times, so with probability 1/3.
If we see -1, we don't know whether it was +1-2 or -1+2, so we have zero bit of the 1 bit of the coin toss. Same for +1. That happens 4 out of 6 times, so with probability of 2/3.

Thus, the expected information to get out of this channel is 1/3·1 + 2/3·0 bit = 1/3 bit, where put in full 1 bit! That's a very destructive additive noise channel.