Make:

Somewhat related, check out the Veritas® Tenon Cutters from Lee Valley Tools.

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Something like would last longer with machine screws rather than deck screws. As the wood dries the head of the screw is going to act like a wedge and cause a split along the screw head. Countersinking might help.

The chair rungs would hold better if they were dried firt, then had a tenon cut with the aforementioned tenon cutter. As the legs shrink around the rung, the rung won't shrink at the same time and it will make a tighter joint. Carving the tenon slightly oblong will make it tighter perpendicular to the grain, and help avoid the same type of splitting as with a screw head. (I didn't figgure this out, it is an old Appalachain chair maker technique)

All of the chairs I have made used dry hardwood and square mortise and tenon joinery with epoxy. They are still rock solid after five years.

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...and if you want to get more advanced, you could use a Beall's Wood Tapping & Threading Kit, also carried by LVT.

Aside: No, I don't work for Lee Valley Tools. I just work two doors down from their HQ and main store. ;-)

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