A wealthy businessman who leads a double life manages to accept his homosexuality only after discovering that his son is struggling with the same conflict.

Protag: wealthy businessman

Antag: businessman’s ‘shame’ of who he is

Conflict: his ‘struggle with who’ he is

Grows: unsuccessfully fighting his demons until halfway story he finds out that he is not alone in this. By excepting what makes his son tick, he must accept what makes him tick.
Inciting incident: It is in the middle of your logline and personally I think that is fine. If you put it upfront it would somehow translate to: “after a homosexual discovers his son also leads a double life,…”

The word “only” has two purposes:

(1) Showing that it takes a lot for the protag to try to really start growing.
(2) Making clear that there isn’t really an inciting – him discovering that his son is gay – incident at the start of your story, but (I would say) halfway.

But if you cut that word, I would still get both those points.

“manages to” is not needed either. Sure “manages to” stand for “the time” his struggle takes. But we as humans know that a struggle always takes time. Then again, if you do not use it, our feeling of “why do we want to see this movie if we already know he’s going to succeed to find ‘inner peace” is going to be too strong, reading the logline. So in this logline construction, I would keep it in.

Also, get rid of the inactive “ing” forms.

Furthermore, you would say ending with “his son struggles with the same conflict”, you just buried the lead. But in this case, exactly this works. Because only by recognizing they are somehow the “same”, they can fight their demons and accept who they are.

A wealthy businessman who leads a double life manages to accept his homosexuality only after discovering that his son is struggling with the same conflict.

could become:

A wealthy businessman who leads a double life manages to accept his homosexuality after he discovers his son struggles with the same conflict.