Let's say you do the simulation K times for the sum of N r.v. samples. You got simulation result quite DIFFERENT from what given by CLT. What should you do?

If N=5 and K=1,000,000, you want use the simulation result.

If N=5 and K=50, you don't know which one should be use.

If N=1000 and K=50, you want just use CLT and throw away the simulation result.

If N=1000 and K=1,000,000, you need check your simulation program -- because this is impossible!