Is 3/16" machine screw equivalent to a #10?
Grainger or Production Tool Supply may have a set with them...
OK, Nobody in my neck of the woods. I'm sure the correct #10 tap can be found in a major city too.
Many of us are limited to Lowes, Home Depot, Ace/True Value, without a journey to the larger town.
If Harbor Freight has number drills, that's news to me. When I needed drills less than 1/16" ( .0625 ) I had to buy them online. Harbor Freight did not have. Nobody in my neck of the woods.
As for this no start thread.. I would first know for sure the start thread of the screw is not the problem. I've had this issue before with cheap mass produced hardware. ell I bought 3 knobs for cabinet doors and on of those screws had NO THREADS, still smooth shank like a clevis pin. Talk about pizzed off, boy was I ever. Anyway... Most times just chamfer that screw end a tad more would do.
If the female thread is the issue I would use the tap in hand to correct the first few threads. That's me though. You do what you feel is best. Good luck with it.