Should the crown be covering the entire tooth?

If the gap you describe is visible and it traps food then the crown restoration has to be re-done. The reason for a full crown placement is to preserve a tooth structure, but in no way it should affect cosmetic appearance let alone have undercuts that trap food. Talk to your dentist or find another one who would be able to address your concerns properly. Good luck, Dr. Daniyar

Whenever possible, I prefer to leave a crown margin about the gums so that the patient can keep it clean. It's also beneficial to leave as much healthy tooth structure as possible. So if the crown needs to be redone in the future, there will be plenty of tooth to place a new crown. If the crown wasn't able to be shaped properly and is trapping food and plaque, that will increase the likelihood that it will need to be redone sooner rather than later. Talk to your dentist about your concerns and possible options.