I feel the following reasons make Shinigami more suited for killing humans than humans themselves:
- Possession of Shinigami Eyes
The name of the person to be killed is an essential piece of information. Humans naturally do not possess the ability to find this crucial detail (Exception: Beyond Birthday from Death Note: Another Note), unlike Shinigami who can see the names of all humans. Of course, humans can obtain this ability as well, but that would require trading half their remaining lifespan for it. And unlike Shinigami, they cannot extend their lifespan to make up for it in any way. This brings me to my next point.
- Ability to Extend Life
By killing a human using a Death Note, Shinigami gain the remaining lifespan of that human. But a human's lifespan remains unaffected by killing another human using the Death Note. By gaining more time to live, Shinigami also gain more time to kill. So what's so special about a longer life? After all, can't humans that die just be replaced by other humans? I will use an analogy to explain why a being with a longer life will be more beneficial for this purpose. Suppose a certain firm sets up a factory to manufacture a particular product that is extremely essential and will always be in demand. Should the firm prefer short term employees or long term employees? If you ask me, I'd say long term employees. Simply because by repeating the process over and over they'd get more efficient at it and also learn to solve any kind of issues that arise. Replacing short term employees will not only require training the new employees, but will also result in slower work and greater chances of error. Short term employees will be preferred in a situation where the manufacturing is only required for a short term or if it is required to introduce innovation in the manufacturing process by application of knowledge from other domains. But humans will most likely live for a long time and there is really no scope for innovation in writing a name on a piece of paper (the method/time of killing is irrelevant to a Shinigami unlike for humans, which I will discuss in the next point). This is why Shinigami with extendable lifespan are preferable.
In addition, a Shinigami getting the remaining lifespan of the humans he/she kills is an incentive to kill. A human cannot extend his/her lifespan by killing another human using a Death Note. So God can't expect an uninterested human to kill for him. On the other hand, even if uninterested, a Shinigami would tend to kill for the mere reason of extending his/her lifespan.
- Indifference to Humans (Although this point is not about the difference in species as per your request.)
A Shinigami living in the Shinigami realm hardly has any chances to be biased towards/against a particular human, or a group thereof. On the other hand, a human residing in the human realm is very likely to have such a bias. This bias coupled with the power of the Death Note can possibly create an imbalance in the human realm. For example, an imbalance could have possibly resulted due to the actions of Kyosuke Higuchi from the Yotsuba Group. If he wasn't caught, Yotsuba might have become a monopoly and eventually infiltrated governments giving a lot of power to a single person. Humans in general would definitely crave for power in the human world. But I don't see a reason for a Shinigami to crave for power in the human world. And a Death Note in the hands of a Shinigami will not give him/her any sort of power in the Shinigami realm. Their indifference to humans and lack of interest in the human realm help them remain unbiased in most cases.
Moreover, possible bias by Shinigami towards humans is even discouraged as Shinigami that kill to save a particular human, or a group thereof, turn into ash and die.
This point is also the reason why giving the powers/abilities mentioned in the above two points to humans will still be an inferior choice. Even if humans get the ability to see the names of all humans and extend their lifespan by killing other humans using the Death Note, the bias will remain, increasing the chances of creating an imbalance in the human realm.