Why this complexity defining AES-GCM counter block? AES-GCM counter block is defined as nonce || IV || counter. Why this complexity? Knowing that it does not need to be secret and its only required property is to be unique, why not use a 128 bits true random number for the whole block? The counter instead of starting at 0 will start at some random value, wrapping around when reaching maximum value.
AES-GCM counter block is defined as nonce || IV || counter


That is not true.  AES-GCM effectively has two different versions.  If you have a 96 bit nonce, then the counter block is (by your terminology) $$nonce || counter$$

However, if the nonce is any other size, it computes a temp value $J = GHASH(K, Nonce)$, and then makes the counter block

$$J \boxplus counter$$

(where $\boxplus$ is addition which ignores the carry from bit 31 to bit 32).

As $J$ is effectively a random number, this second possibility is mostly what you are suggesting.

Now, it has since been found to have inferior security properties compared to the first idea (the proof had a flaw), so I don't suggest you actually use it