So did Abraham Lincoln and Winston Churchill. I think that Eagleton's elimination was not so much because of the earlier depression but because of the electroshock treatments.
Churchill suffered badly from depression, but was otherwise the best his nation had for the job at hand in those times.
Lincoln’s “depression” is a matter of debate.
Eagleton’s political demise was, as you stated, the result of the electroshock rather than the depression itself.
In Eagleton’s day, anti-depressants were crude and riddled with side effects. Even with today’s sophisticated anti-depressants, I still would not want a POTUS with depression issues.