You are assuming that all contract labor paid $600 or more requires a 1099-NEC.  Your assumption may not be correct.  If the contractor is incorporated a 1099 is not required to be issued. 

Fill out the Schedule C whichever way makes the most sense to you.  As long as the correct amount of net income is being reported and the correct amount of income tax is being remitted you are ok. 

Inform the client regarding their obligation to file information returns when required and the possible consequences if they do not comply.  Its best practice to cooperate with your government.