Why XOFs are more convenient than Hash Functions in modeling Random Oracles
Suppose you are instantiating a cryptosystem, such as RSA-FDH signatures with 2048-bit modulus, or EdDSA over a $b$-bit curve for $b \gg 256$ like edwards448, which needs a sizable hash function modeled by a random oracle.  But the hash functions available at your disposal are only SHA-2.


EdDSA needs a $2b$-bit hash function.  For edwards25519, it was enough to use SHA-512, but not for edwards448.
RSA-FDH with a 2048-bit modulus needs a ${\approx}2048$-bit hash function.  None of the SHA-2 hash functions are big enough for this.


What do you do?  In order to instantiate one of these cryptosystems, you have to pick some way to compose the SHA-2 functions into a larger-output hash function.  It's not a priori obvious what instantiation to choose.


One might naively choose $H(x) \mathbin\| H(H(x)) \mathbin\| \dotsb \mathbin\| H^n(x)$, but this is manifestly not indifferentiable from a random oracle.  (You and I might not choose this, but I have seen people reach for it in practice.)
One might choose $H(x \mathbin\| 1) \mathbin\| H(x \mathbin\| 2) \mathbin\| \dotsb \mathbin\| H(x \mathbin\| n)$; of course, one must also choose how wide the numbers are, how they're padded, etc.
One might choose $H(H(x) \mathbin\| 1) \mathbin\| H(H(x) \mathbin\| 2) \mathbin\| \dotsb \mathbin\| H(H(x) \mathbin\| n)$ like you suggested, but it's not clear what purpose the extra $H(x)$ serves.


Personally I would choose option (2) with little-endian 64-bit integers, but justifying that choice requires further thought and explanation, and not everyone will make the same choice.  With a standard XOF there is no need to think about these questions: you just ask SHAKE128 for as long an output as you want.

This is part of why Ed448 was designed not to use SHA3-512 (in some counter mode or whatever) by analogy with Ed25519's use of SHA-512, but instead uses SHAKE256 with an appropriately long output.  The complexity of the OAEP standard arises in part because there were no XOFs available so one had to be invented (the ‘MGF’); it would have been much simpler and easier to understand if XOFs were available.