What is the surest way to permanently kill a junk tree besides cutting a full ring around the trunk's bark?
Probably the best way to remove a tree is to cut it down and remove the stump. Herbicides will kill the tree, but you will still need to remove it. Depending on the herbicide, you may affect other plantings nearby or kill other trees at the same time.
There are several types of herbicide to eliminate a tree, you just need to be careful as some will be soil active, meaning it will stay there and kill other vegetation too. Be warned, some trees may be attached to the root systems of other trees(locust is an example), and if you treat one, others around it can die too If you can get the tree down, flush off the stump and use a herbicide around the outer cambium layer. This is the living tissue just inside the bark, and you don't need to treat the whole top. Look at the labels in the store and look for the active ingredients tryclopyr or picloram. Usually you can buy it off the shelf, since it is non restricted. These are in many of the brand names you can get at a home store. If you can't get the tree down, you can treat the outside, spraying from knee height down to and including the root flare. .
I don't think I've ever run across a tree in the field that hasn't formed a root graft if a related tree species (ie red oak to red oak or white oak sometimes) is within root grafting distance. To completely kill the stump with Roundup you are going to have to remove the bark from the outer cambium because as long as that tissue is protect it will get the chance to grow back. Removing the bark with dry out the cambium. The herbicide Roundup is typically your best bet because it's pretty nasty stuff. The problem with just spraying the top and surrounding area is some trees are very persistent and will easily shoot up suckers (a new baby tree but grown from the root stock which develops weaker wood) in an attempt to survive. Maples, Aspens, and Willows are notorious for doing this. Roundup is typically the most common herbicide used by tree companies because the main active ingredient of roundup is the isopropylamine salt of glyphosate. Drill holes in the top of the trunk and pour some roundup in the holes, this will aid in killing the roots to prevent suckers. Roundup however is toxic to wildlife just as a heads up BUT it does not sterilize the ground like other herbicides would which would prevent you from being able to plant anything else sometimes up to several years. When you are using an herbicide properly, as in you're not overdosing, root grafts shouldn't really be much of a concern because it would take a lot of herbicide to cause any major damage. As I said herbicide damage to a neighboring tree is unlikely if the correct dosing is done but if you start to see signs of dieback (dead branches that weren't dead before), Chlorosis (yellowing of the leaves, or Scorch (literally looks like the leaves have been set on fire) then you have poisoned the nearby tree.
My uncle would cut a tree down to grown level, add charcoal and let it burn the stump and its roots away. However, you can end up with a sunken area in the yard and possible fire damage if it spreads.
Probably the best solution is to remove the stump by hiring a tree company to grind out the stump, or rent a stump grinder. This can get pricey though.