Efficiency is one reason why e.g. GCM as a MAC would be nice. GCM uses GMAC, which can be a relatively fast operation with hardware support. Modern (AMD 64 compatible) processors have an Intel defined multiplication instruction called
PCLMULQDQ to support it (Intel hosted PDF).
Commonly GCM is said to have 1.5 passes instead of the actual two to indicate that the MAC pass is faster than what you would commonly expect (the speed depends on the implementation and isn't necessarily 0.5 times the speed of a "normal" MAC). However, in software it could actually be slower than other MAC implementations.
Note that if the ciphertext is empty that the GCM spec will simply hash the padded AAD, the length of the AAD (64 bits) and the length of the ciphertext (64 bits set to zero). As the lengths together make one 128 bit block, this is relatively efficient. The computation is just GMAC according to the specification. Some libraries still have a separate GMAC implementation though.
CPU cycles on Intel
Currently GMAC scores about 2.33 cycles / byte using the optimized instruction, SHA-256 in software somewhat over 11 cycles / byte but in hardware (Intel SHA extensions), it may be faster at 1.9 cycles / byte (!). Beware that these are optimal results, it's not required that software makes use of these instructions. For smaller messages you should also take in mind that there will be a non-negligible overhead for both GMAC and HMAC constructions.
Above are general cycle counts on modern 2017-2020 Intel CPU's. You should only use these as numbers that give an idea of the overall performance.
And yes, the nonce should supplied and the key / nonce combination should be unique. Personally I would not use GMAC for MAC only; it is a much more risky algorithm compared to e.g. HMAC. Please make sure your implementation will remain secure if you do use it.
Other AEAD ciphers
A similar argument can be made for Poly1305; it is probably somewhat faster than GMAC in software.
Other AEAD constructs such as CCM and EAX are build on more common MAC constructions. CCM uses CBC-MAC while EAX uses (AES-)CMAC. Both are so called two pass protocols, so using those as MAC will not offer any performance benefits. Instead of using CCM or EAX as MAC it is probably best to just use HMAC or indeed AES-CMAC directly, whichever is best for your particular application and system.