If your TV has a game mode, definitely start with that as it will help tremendously. My brother has an OLED with game mode and we tried playing MLB without using game mode once and I don't see how anyone could hit anything online just on a regular TV. At least I know I couldn't anyways. As for your HDR settings, I have mine turned off in my xbox settings since MLB is the only game I play on there, however I'm not real sure how helpful turning it off has been. I'm not sure of any other settings that are going to help but if you are able to hardwire your xbox or playstation with an ethernet cable instead of using wifi that will also help.
If you already have your TV in game mode and you are on the fringe of making world series, then yes, a monitor could definitely help you get over the hump. However, if you have it in game mode already and are struggling on All Star then definitely don't go into it with the expectation that a monitor will get you to World Series right away.