If a Kalman filter can only receive information on $(x, y)$ position, is there any reason to have acceleration as part of the model?
Whether to include a particular parameter in the model depends on how much you know about the parameter. If you're already including 2D velocity in the state, then the acceleration will be coming in through the random noise assumptions you are making.

Whether tracking a person will improve if you include the 2D acceleration in the state depends on what assumptions you can make about how the person's acceleration changes.

For some models, we might know that the jerk$^1$ is completely random but not the acceleration. In that case, it would make sense to include the acceleration and feed the randomness in as the jerk (change to the acceleration).

${\tiny^1\sf \mbox{The derivative of acceleration, I'm not casting aspersions. }}$