After using this system for several sessions and many mock tests, I am pleased to say it works exactly as advertised for my group. Combats are faster. There's less confusion about positioning. I don't have turns slowed down as players count out their movement trying to avoid OA's. AoE spells are a lot easier to handle, faster, and more effective.
As the DM I have enjoyed the benefits immensely, and it has actually helped to mitigate some elements I always found difficult to manage without making any player feel ganged up on. With Grid play, ranged type characters (predominantly archers and mages in my games) enjoyed a hefty advantage of always been far from the fight. Even though I include enemies with ranged attacks and magic users in near every combat, it is still very easy to use a grid to effectively deny line of sight on most turns. Lost of stuff to hide behind in most battlefields. In order to present harder challenges, I would need to make almost every threat they faced be highly mobile or have powerful ranged abilities, which if you look in the monster manual many creatures just do not have.
By using this TotM system, I can have enemies break off and rush any character in the heat of battle. There's defender types who Intercept to protect their archer allies, but there's only so many they can bog down. It has helped to make even swarms of weaker enemies feel more threatening. A mob of goblins ordered by a hobgoblin leader to take out the "finger wiggler" all rush around the front line, and a few still get through to harass the wizard.
I admit some of my players didn't like the extra attention at first, but they quickly got used to it, and even started looking at their abilities and spells to plan out defensive strategies.
I'm sure a better DM may have been able to make things challenging even using Grid, but at least for myself, the Roshambo-Style TotM is now my go to.
(Plus, I suck at drawing maps anyway, and there's less need using TotM, so it's a win/win!)