Why are the same set of controls used for both OLS and 2SLS? I have been reading several papers that use 2SLS. These papers usually also show the OLS regression results. I noticed that when authors report results for 2SLS, they also include control variables that were used in the OLS regression.

If 2SLS is supposed to solve the omitted variable bias, couldn't we omit the control variables that are included in OLS because we do not have to worry about omitted variable bias? For what reason would authors include those variables?
The reason for that is that 2SLS does not solve omitted variable bias without its cost.

First, for every omitted variable you would have to find suitable instrument. Finding instruments is incredibly hard so why would you purposely create more omitted variable bias? Unless you are trying to challenge yourself for fun or doing it as a practice exercise for students it does not make sense.

Second, actually 2SLS still has bias - the advantage of 2SLS is that bias decreases as strength of the instrument increases, so with good instruments 2SLS is better than OLS in terms of bias. But in practical applications you rarely have perfect instruments- so even in case you could feasibility instrument the other variables you omit on purpose you will be just adding to the bias of your model.

Hence you should always include all relevant controls that you can include