Damn, I've been trying to think of a charity I could start, broom never occurred to me.
Looked at the broom problem a bit a few years ago. A big problem in Australia and New Zealand too. And there they treat it as a serious problem.
Cutting might be quick and give a sense of accomplishment, but is not effective in eradicating the plant. It will however at least stop that years seeds from forming.
The roots have to come out. There are a few tools on the market, eg,
http://www.pullerbear.com/ ,
http://www.extractigator.com/ ,made here on the island that will get all or part of the root.
Interesting side fact on broom, seed can lie dormant for 30 years before sprouting.
Wish them luck, but chemicals are the only solution I can see working, and it, the field, would have to be monitored and repeated if necessary over several years.