Here is how I understand it - The company Tresona was created by a few lawyers to help music publishers gather payments from groups using music that they hold the copyrights to. They routinely have a team of workers that scour the internet (Facebook , YouTube, Vimeo, etc.) looking for videos of performances (marching bands, show choirs, indoor drum lines - all groups that would potentially be performing arrangements of previously published pieces of music). When they find videos, they check with the publishers to see if permission to arrange was granted to the group. If permission was not granted, Tresona's lawyers send nasty letters to the administration (head principal and head superintendent) and have even filed lawsuits against groups for copyright infringement. One of their first such suits was against the Burbank (Cal) high school show choir program that the tv series Glee was based on. Check here for an article.
To use your words, Tresona has a "stranglehold" on the process as they have shown the music publishers that they (Tresona) will do all the legwork and chase down potential copyright violators and therefore can bring revenue to the publishers.
Tresona then took their "knowledge" of marching band and such and created a "service" where groups can go through them to secure permission to arrange pieces of music. The kicker (of course) is that Tresona kindly adds an additional fee (usually a couple hundred dollars) on top of whatever amount the music publisher wants to give permission.
You ask "why not go about things a different way"? The "different way" would involve directors contacting the music publishing companies directly to chase down permission to arrange directly from them. Sadly, many publishers are starting to direct all such inquiries to Tresona directly.