Did he do impersonations of his father?
Incorporate his father into an act or performance in some way?
Is Anthony in the entertainment business?
Folly:
Did he do impersonations of his father? NO
Incorporate his father into an act or performance in some way? NO
Is Anthony in the entertainment business? NO
Did his father’s death cause any physical injury to Anthony?
Did his father’s death cause any physical injury to Anthony? NO
Was he responsible for his father’s death?
Was he not responsible, but suspected of being so?
Was he responsible for his father’s death? NO
Was he not responsible, but suspected of being so? NO
Did the manner or circumstances of the father’s death matter?
Did the father’s death result in another something happening and the another something was the cause of the near career wrecking?
Is Anthony or the father famous?
Was Anthony’s livelihood based upon his father’s reputation?
Folly:
Did the manner or circumstances of the father’s death matter? NO
Did the father’s death result in another something happening and the another something was the cause of the near career wrecking? YES (good question, and I’m glad you asked it!)
Is Anthony or the father famous? ‘Famous’ is maybe too strong a word. ‘Fairly well known’ is a potential YES for both, depending on who you’re talking to
needscoffee:
Was Anthony’s livelihood based upon his father’s reputation? NO
Is Anthony more commonly called Tony?
Is Anthony more commonly called Tony? YES
Was the near-career-wrecking related to:
Anthony being physically unable to do his job?
Mentally unable?
Broader social conditions, such as high unemployment in the industry where he worked?
Other people being unwilling to employ or do business with Anthony?
… as a result of a scandal of some sort?
WAG: Did Anthony have a secret that he didn’t want to get out? If so, was it published (or inferrable from information published) in his father’s obituary?
Another WAG: Did Anthony have the same name as his father? If so, did people mistakenly think HE was dead?
Was the near-career-wrecking related to:
Anthony being physically unable to do his job? NO
Mentally unable? NO
Broader social conditions, such as high unemployment in the industry where he worked? NO
Other people being unwilling to employ or do business with Anthony? YES
… as a result of a scandal of some sort? NO
WAG: Did Anthony have a secret that he didn’t want to get out? If so, was it published (or inferrable from information published) in his father’s obituary? NO
NO to both of these
Were people unwilling to work with Anthony…
Because of something Anthony did? If yes, did he do it because his father died?
Because of something he didn’t do, but other people mistakenly thought he did?
Because of some person he associated with?
Because he belonged, or was thought to belong, to a socially stigmatized group? (E.g., gay, communist, ex-convict, etc.)
Because of some fact about him that isn’t usually stigmatized?
Was Anthony in a profession where public image is important, such as an entertainer or politician?
Would it be helpful to know the historical period when this happens?
Would it be helpful to know the country?
Were people unwilling to work with Anthony…
Because of something Anthony did? If yes, did he do it because his father died? NO
Because of something he didn’t do, but other people mistakenly thought he did? NO
Because of some person he associated with? NO
Because he belonged, or was thought to belong, to a socially stigmatized group? (E.g., gay, communist, ex-convict, etc.) NO
Because of some fact about him that isn’t usually stigmatized? YES
Was Anthony in a profession where public image is important, such as an entertainer or politician? YES
Would it be helpful to know the historical period when this happens? Maybe
Would it be helpful to know the country? YES
Just to save a bunch of narrowing-down, the period is mid-Twentieth Century, and the country is the UK
Tony Hancock?
Tony Benn?
It is indeed Tony Benn! I have no idea what relevant thing might have happened to Tony Hancock though … do you know the incident I was thinking of already or were you just guessing famous UK Tonys? 
In any case, the full story is on his Wiki article, so - details:
Tony Benn was a prominent Labour MP whose father William Benn had also been a prominent Liberal-then-Labour MP, and been granted a peerage. It was never intended that Tony would need to inherit the peerage, since this was an automatic disqualification from his House of Commons seat - but his older brother died in WWII.
When Tony’s father died himself, in 1960, Tony was removed from his seat, being now a peer of the realm, and ineligible. He spent two years fighting for the right to renounce a peerage, which he finally obtained, was voted in again, and had a long successful ministerial career after that.
Just naming famous Tonys. (Tonies?)
Tony Hancock struggled with depression all through his life. I did wonder if maybe he had a particularly bad time after the death of his father.
That was an unexpected way to finish the puzzle! But, y’know, any way you get the skin off the bear’s a good way