Most likely the V diameter is too small to allow the large areas to be cut.
Select the counters, inset them by half the V diameter twice, then select the inner and outer paths for a V carve, and use the middle one for a pocket — you’ll need to clear some material with a chisel if there’s a sharp corner.
Alternately, use a tool such as Vectric Vcarve which can depth limit Vcarving.
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