Stretching lead came is so ingrained into the literature and general thought that it is difficult to regain the purpose of the practice. But I will try.
The purpose is to straighten the came
The purpose of putting the lead into a clamp and pulling on the other end is to straighten the lead came. It is much easier to work with a straight came than one that is curved or kinked. It gives visually straight lines, it provides smooth and sinuous curves without interruption in the line of the curve.
It is said that some came is “pre-stretched”. This is really the result of alloys contained in some lead to make it stiffer. It still needs to be straightened before use. If the lead came is already straight, you do not need to do anything else before using it. If you drop or otherwise accidentally bend the came, you need to straighten it before continuing.
Stretching can weaken the came
Pulling on the lead came is not to stretch it, it is to straighten it. Stretching the lead can make it weaker. Lead drawn beyond its structural limits will break. But you can weaken it before the break. You can test for this weakening of the came by observation. If you see "alligator" marks on the surface, you have weakened the came by putting too much effort into the pull. Straightening the lead must avoid so much force as to weaken the structure of the material.
Straightening not Stretching
The amount of effort to be put into straightening the lead came is just enough to make it straight. This will vary depending on how straight the came is at the start. The reason for drawing the lead toward yourself is that you can see as you look down the length when the lead came is straight. If you are pulling vertically, it is more difficult to see when the lead becomes straight.
If the lead is badly kinked or twisted, it may be best to cut that section out. If you continue to pull to straighten a difficult section, you can weaken the whole length of came. First, ease the kinks and twists out as much as you can by hand. Then do an initial straightening pull. This initial straightening pull will show where the problem(s) lie. You can cut that section out and straighten the remaining pieces without stretching the lead to the point of weakness.
Of course, you must employ some basic safety rules. Make sure the lead is securely clamped. In the cleat style lead vices, you can give the lever a thump with the pliers to ensure the teeth are set into the lead before pulling on the other end. Other vices need to have other ways to ensure that the end is held securely.
The other basic safety rule is that you should brace yourself against any break of the lead, or slip from the vice. One foot should be placed behind you so that in case of breaks or slips you will not overbalance and fall. This has the added advantage of ensuring you cannot put your body weight into the straightening effort.
There are other common sense rules, such as gloves, removing obstructions behind you, etc.
Remember that the purpose is to straighten, not stretch the lead came.
If you are putting your foot on the bench to add force to the puilling of the lead in a vice on the bench, you are putting too much effort into the job and risk falling when the came breaks or slips out of the vice. If your whole body weight is being used to draw the lead toward you, you are using too much force. If you can see signs of a pattern developing on the surface of the lead, you are using too much force.
Straightening the came is not an exercise in a workout programme. It is a steady firm drawing force until the came is straight. If you have to use more than usual force, stop and figure out why. Cut out the difficult section so you do not weaken the came. Then straighten the remainder and continue leading.