Half lap on the legs, lengthen the stretcher and set it in the V formed by the legs. Ends of the stretcher can be beveled and glued and doweled in place. Much stronger than tenon in the half lap. Half lap joint can be moved down if needed.
I'm not sure I would agree with you here.
There is no way a pegged tenon joint would not be stronger than what you are suggesting.
Even doweling and gluing the stretcher as you described would be a weaker joint, especially on the top faces of the stretcher.
Not to mention the difficulties in trying to relocate the half lap joint. That needs to be at the meeting point of the two cross assemblies,and cannot be moved.
Also, why bevel the ends of the stretcher? That just further weakens the joint you are proposing.
As stated earlier, the pegged tenon provides a great deal of strength to not only the stretcher, but also the half lap joint.
Once pegged, and the wedge is driven in place, this joint would be much stronger than the wood itself, and mechanically prevent it from coming apart without physical damage to all 3 parts.
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