That would not work.
Let's say regressors are $x_1$ and $x_2$. $x_1$ is endogenous, and $x_2$ is exogenous control. You're interested in causal inference for $x_1$. What you're proposing is to instrument $x_1$ with $z = x_2$.
The proposed 2SLS procedure is then the following.
In the first stage, you would regress $x_1$ on $z$ and $x_2$ to get $\hat{x}_1$ and regress $x_2$ on $z$ and $x_2$ to get $\hat{x}_2 = x_2$.
In this case, the regression of $x_1$ on $z$ and $x_2$ would be trivially multi-colinear, and $\hat{x}_1$ is just given by regressing $x_1$ on $x_2$---i.e. $\hat{x}_1$ is a scalar multiple $x_2$.
So the second stage regression, where you normally regress $y$ on $\hat{x}_1$ and $\hat{x}_2$, is again trivially multi-colinear. You would be regressing $y$ on only $x_2$---you have lost $x_1$, the regressor you're interested in, completely.
Empirically speaking, a variable cannot serve both as control and an instrument.
An instrument $z$ channels its exogenous variation through its correlation with $x_1$ (notice this statement contains both conditions for a valid instrument).
Now if you have a control $x_2$ in the regression, then $z$ must have some residual variation after controlling for $x_2$. Obviously, $x_2$ has no variation after controlling for $x_2$. This is the problem.
(Even more informally, what you're proposing would be a universal solution for finding instruments. Clearly that cannot be the case.)