My first though upon reading your post above was to go look for Benson's Probate documents (Will/Succession). I found the documents online (attached in .pdf format).
Tom Benson's Will and Succession docs are a bit perplexing to me. as IMHO it is strangely written and leaves some questions in regard to what you've stated. Tom Benson left his estate to the "Trustees" of the Tom Benson July 2015 Revocable Trust. Who are these Trustees? While Gayle is entitled to all of the income and has sole voting rights for over the Saints and Pelicans entities, she was named Co-Executor along with Dennis Lauscha of said Trust, and Lauscha was named Executor of the Succession, with Gayle being named Succession Executrix. Heck, even Loomis and Bensel are named as potential Succession Co-Executors in the latter should Lauscha and then Gayle Benson either are unable, or elect not to, take on the responsibility.
So, what all, if anything, does sole voting rights encompass where the day-to day operations and management of the teams are concerned?
There is a lot more to it, however to me, it appears that Lauscha, while not having voting rights, may have some autonomy in being in charge of the day-to-day operations, while Gayle collects checks and votes on league related issues and any decisions on whether to sell either team, etc. (I'm not sure about that), which would basically lend credence to your statement quoted above, however I may be misinterpreting. I'm not an Attorney. So, hopefully someone that is an expert in Descent and Distribution in the State of Louisiana can interpret it all correctly for us.