A lot of horses will use the reins or a tie down as a "third leg" if you will. They kind of balance off it, which is what it sounds like your polo pony is doing. At 19 years old I would say she's probably very used to it and would miss it if you chose not to ride in it.
Any training device has the potential to become a crutch to a horse or it's rider. Even the most skilled riders can jam themselves by becoming too reliant on a specific device, that's why, if possible, it's best to learn to work through whatever issue without it.
That being said, I'd say your horse is probably a "chain-smoker" by now and feels very comfortable in the tie-down.
"Be a best friend, tell the truth, and overuse I love you
Go to work, do your best, don't outsmart your common sense
Never let your prayin knees get lazy
And love like crazy"