The definition of authenticated encryption is, essentially, indistinguishability of the ciphertext plus MAC security of the tag. You're right, the tag does not need to be indistinguishable from random, but in the majority of the cases it is anyway.
One significant difference between these non-cryptographic generators and cryptographic ciphers is that of latency---generators often optimize for latency, whereas ciphers optimize for throughput. By taking a cipher and a sufficiently large buffer, you can have low latency as well, at the cost of some memory.
For comparison, consider MORUS, AEGIS, and Tiaoxin, three unbroken contestants of the CAESAR competition. MORUS uses only AND, XOR, and bitwise rotation, and achieves somewhere between 0.5 and 0.66 cycles per byte on current x86 chips. AEGIS and Tiaoxin use the AES round, and where AES-NI is available, performs at between 0.15 to 0.25 cycles per byte. The claim somewhere above is that PCG can do 0.25 cycles per byte (or 4 words per cycle) when going all out with AVX-512; that's only hard to beat if you've not been paying attention.