That does indeed describe a properly engineered 50 amp device.
It is normal for appliances to use slightly lighter gauge wire than would be used in building wiring. You can observe that inside electric ranges, dryers, and water heaters, where you will see the 10 or 8 gauge incoming supply wires to a terminal and then 12 or 14 gauge wire going from the terminal block off to the components.
Because the wire is inside an enclosed box, and is not tightly bundled or surrounded by thermal insulating materials, its elevated temperature will not cause any danger. It will run a little warm at peak current, but be far below "hot". The same is not acceptable inside dwelling walls were it will be in contact with wood, fiberglass, cellulose insulation, etc.
As for standards, I have determined this is beyond the scope of the NEC and falls into the domain of the National Fire Protection Association (NFPA) standards. That seems to cost $$ to view.
This is a fairly "common" issue, that typically occurs due to high resistance at the terminal. The high resistance can be caused by corrosion, or a loose connection. The high resistance causes heating, which causes higher resistance, which leads to more heating. Eventually the insulation on the wire starts to melt, and/or burn. Usually the heat will transfer through the terminal block, and into the smaller wiring in the stove. When the heat becomes too much, the smaller appliance wiring will fail and open the circuit.
In most cases this is not a super dangerous event, as the wiring is contained within the stove. However, there is a chance that this can lead to a fire in the house, especially if the cover is not properly installed.
Causes
Corrosion
Corrosion is more commonly a problem if the wiring is aluminium, though can still be a problem with copper wiring. Because stoves/cooktops/ranges draw large amounts of current, there is typically some heating of the wiring during operation. The constant heating and cooling cycles of the wiring can lead to corrosion, and/or hardening of the wiring over time. This corrosion and/or hardening increases the resistance of the wiring, which cause the heating to be worse.
Solution
Occasional inspection of the terminal block and wiring, can prevent the corrosion/hardening from getting too bad. If you notice any discoloration or corrosion, have the cord replaced. There are also conductive pastes available, that can be applied to the wiring and terminals to help prevent corrosion. While this should always be used with aluminium wiring, it may also be applied to copper wiring.
Loose Connections
This can be caused initially by not tightening the terminals to the proper torque (usually specified by the manufacturer), or over time by heating/cooling cycles. Loose or bad connections can cause arcing and/or higher resistance. As discussed before, high resistance leads to heat.
Solution
Make sure all terminals are tightened to the manufacturers specifications. It's also a good idea to tighten the terminals after a few hours of operation. As stated before, occasional inspection can also prevent this situation.
Why didn't the breaker trip?
The heating was localized to the stove\cooktop\range, so the thermal protection in the breaker would not trip. Since stoves\cooktops\ranges draw large amounts of current during normal operation, the current during the event likely didn't go high enough to cause heating in the breaker to trigger the thermal protection. It could also be that the breaker is faulty, so it should be inspected.
Is the stove scrap?
The stove may be salvageable, though you'll want to thoroughly inspect all the wiring to insure it is not damaged. Replace any damaged wiring with equivalent parts (cooking appliances use high heat resistant wiring).
If the stove is still under warranty, you'll want to contact the manufacturer to discuss your options.
Best Answer
The #10 wire is too small coming from a 50 amp breaker. #10 or 10awg wire is only rated for 30 amps by code - this is why someone advised you to change to #6.
Just to make sure we are talking about the same wire, there should be #6 or #8 coming from the breaker to the range. If the wire that burned off is inside the range going to let's say an element or burner, then stay with the same size wire (high temp fixture wire). Since you are talking about #14 I am guessing it goes to an element. The important thing about connections at elements is that you use connectors rated for high temp, for example many oven elements are male spade disconnect type connectors. Using a standard female crimp connector will have you back in there because the standard ones can not handle the heat. The ones I use on ovens are rated for 900f. So when replacing the wire if crimp connectors are used at the element use high temp ones or splice a piece on the existing wire using a high temp wire nut.