What kind of attack could one perform if you generate private keys using SHA3(master_key || nonce)? I'm aware HMAC is more appropriate than SHA3(master_key || nonce), but I do not understand what kind of attack could be performed against that strategy. How could an attacker use this fact to figure out your private keys?
KDFs

What you are trying to describe are different implementations of a key based key derivation function (KBKDF) which in turn is a specific form of key derivation function (KDF) that uses a key (and not a password).

The attacker would not be able to reverse the hash when using SHA3(master_key || nonce). The reversal could not even happen for SHA-1; the fact that SHA-1 is broken for collision resistance does not matter for this construction. HMAC is more appropriate because a keyed hash / PRF has better cryptographic properties than a hash.

$H(k | c)$ is actually a known KDF called KDF1 or 2 (those just differ in initial counter value $c$). As described here it is probably still secure. However, if you want real, standardized, proven security I'd probably recommend HKDF using SHA-512 as underlying hash. That should be secure until SHA-512 is broken to the extreme.

On HMAC

HMAC protects the underlying hash function against certain attacks. It should always be secure if a large enough key is used and the underlying hash is sufficiently secure - but that's even the case for MD5. HMAC also alleviates length extension attacks for the underlying hash. However, for SHA-3 there should be no length extension attacks possible, so SHA3(master_key || nonce) would not be affected by length extension attacks anyway.

HMAC is rather overkill for SHA-3 as it hashes the key twice after applying two different masks, while the protection it offers is inconsequential to SHA-3. So HMAC-SHA3 could considered an option because the HMAC construction is secure with any sufficiently secure hash. The more efficient KMAC construction is described in SP 800-185 and could be used instead - if it is available in the programming environment anyway.

TL;DR

Now that you know that you need a KBKDF, I suppose it is a good idea to use a well described one. I'd strongly recommend HKDF or HKDF-expand. NIST SP 800-108 has a few more KBKDF's, but NIST standardized HKDF afterwards for good reasons.