Should I Eat my Cover Crop?

I've been developing an interest in keeping cover crops in the ground over Winter as a method of reducing soil erosion and improving fertility, but I've heard conflicting reports that a cover crop should be mowed down before the plants can mature in Spring otherwise the benefits of nitrogen fixation in the legume plants will be negated by the energy the plant uses to produce seed. I've never intentionally left a cover crop in my garden over Winter intil this year so I have no practical experience to confirm whether or not it makes a difference if you allow the cover crop to fully mature to seed. Perhaps some more experienced members on this forum can enlighten me on the area of cover crops.

Secretly, I've also been hoping to increase my food yield per acre by rotating a warm season crop in a given bed during late Spring through early Summer (e.g. Corn, Sorghum, beans, cowpeas, squash) and then planting a Winter crop in the same bed for harvest the following spring (e.g rye, winter wheat, vetch, austrian peas, parsnips). If I were to grow the cover crop to maturity and harvest the crop at maturity the following Spring, would I be defeating the purpose of a cover crop?

Depends on your cover crop goal. As you mentioned adding nitrogen rich organic matter is usually with terminating and incorporating.  But your goal of producing an overwintering crop is very worthy. Having roots feed the soil with exudates over the winter may be all your soil needs. In my case winter wheat grows all winter and produces chicken feed in early summer. The straw provide mulch and the roots can be left in the ground when transplanting squash and pumpkins.  Winter hardy kale feeds me all winter and then a few of the strongest are left to produce seed for the next year.  Good to reach out for others observations. Add your observations, make a plan, implement it then adjust the plan based on the results. Share the results on the forum.