Putting your two queries together, I would say, DON'T use the strimmer on the marestail! That would just distribute the bits of black root around, and you'd get new plants from each of them.
I don't know how well the flame burner would work, as I've never tried it, but I do know that you have to get rid of the roots as well as the surface growth, or it will just come back. We have a rockery which is all rocks with pockets of earth in (in contrast to a lot of rockeries which are a pile of earth with some rocks doted about on.) It took stripping out all the plants and then years of weedkillers twice a year to kill it off, and even then the little spiky shoots would peek up occasionally and have to get individual treatment, spraying something into a plastic bag that would be absorbed down to the root, bruising the shiny grren surface of the marestail and trapping it in the bag in contact with the weedkiller, then sealing the bag and leaving it to die back. The surface is so shiny that the weedkiller just runs off unless you bruise it.
It reproduces by spores as in the picture, which are on an unobtrusive brown spike which appears sneakily in March or so, before the green shoots. ~It is the same colour os the ground, so easily missed. If you spot one, wrap a poly bag over it when you remove it, so that the spores don't fall in the process.
Marestail is a plant that has been around for millions of years and survived, so it is very tough! Wikipedia says "Equisetum is a "living fossil" as it is the only living genus of the entire class Equisetopsida, which for over one hundred million years was much more diverse and dominated the understory of late Paleozoic forests. Some Equisetopsida were large trees reaching to 30 meters tall."
en.wikipedia.org/wiki/Equisetum#Description
