In humility, can a somewhat large penis be the reason it never is fully rigid during sex?  

Any size gets rigid. Any size of penis should be able to get rigid. Reasons for semi-rigid penises include medical problems, psychological factors, medication or drug effects, past penile trauma, excessive recent sexual activity, older age, fatigue, hormonal imbalances, etc... A primary care doctor or a urologist can evaluate and figure out what the issues might be.