It is illogical that an old man would need a walking stick when he is as agile as a circus monkey.
Just watch Yoda fight Count Dooku. Why does he need a walking stick?
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Essentially, it's because he is an "old man" by nature. His speed and agility during combat are the result of using the force. Ataru, Form IV and the lightsaber form that Yoda canonically used in Legends canon, uses the force to power that speed and athleticism.
By allowing the Force to flow throughout their body, they could overcome their physical limitations—including old age, as was the case with Master Yoda—and perform amazing feats of acrobatics, such as somersaults and backflips, not only for attack, but also to evade the slashes and strikes of their opponents.
Disney canon has not, to my knowledge, explained it, but the fact that Yoda seems frail and leans on his walking stick both times he finishes combat suggests that his ability to power that athleticism is limited, and likely takes a toll on his body.
Yoda's cane is not merely a walking stick. It is a Gimer Stick that has purpose beyond a simple mobility aid.
Gimer Juice may be sucked or chewed from the bark of the stick, providing Yoda with nutrients and a form of anaesthetic. The juice can be used as a substitute for food and the anaesthetic calms the senses, aiding in meditation as well as numbing pain.
Yoda isn't hobbling around, leaning on a stick because he's old and in pain. He's hobbling around, leaning on a stick because the stick gets him high and he can't walk straight without it.
This information comes from Star Wars: Absolutely Everything You Need to Know, a book considered part of the Expanded Universe by Disney and is therefore only Legends canon. Having said that, information from the book that is consistent with the canon universe is considered canon as far as the Holocron Continuity Database is concerned.