I don't regard them as being the same at all, but since you do, let us move forward with the analogy: At some point, the persons legs will heal and they will undergo physical therapy to help them walk again. Is that not the same as walking it off?
The beauty of a "mental health" issue like depression is that you can control exactly when that healing begins.
First, wow, you literally have absolutely no clue at all what you are talking about.
Second, a broken leg doesn't just heal, it needs fixing first, it needs medical intervention. Broken legs before medicine and without the knowledge to reset the bone led to death. Even resetting the bone in many cases isn't enough to heal fully hence a lot of broken bones are operated on and held in place with screws so that is does heal correctly.
You can't choose to control when healing begins, the brain is infinitely more complex than a broken bone and while we have an almost complete understanding on how to mend broken bones we haven't obtained anywhere near the level of knowledge required to fix mental health issues. Depression itself can prevent you via it's symptoms from seeking health. The attitude of ridiculous uninformed people like you have for decades prevented people from seeking help as well.
Most people who go the pill route end up trying many combinations of pills over the course of potentially decades before they find one that works for them, largely because brain chemistry is incredibly complex and how a drug affects people changes from one person to the next. Some of the pills make people MUCH worse, many of the pills have significant side effects.
When you don't know what in the hell you're talking about, and make no mistake you haven't got the slightest clue, running around giving awful advice is harmful, not helpful.