In English, I would like: cannot get the gate to open (yours might be understood as, my dad cannot get permission to open the gate since he is treated as a child). I got 'wrong' for "..doesn't manage to open the gate."
I always had the idea conseguir is more 'to succeed', while poder is really about ability. So you would never use conseguir when you ask people to do something. But I'm not one hundred percent sure.
Spanish and Portuguese are cognate languages. In Spanish we have the verbs conseguir and poder. The former is equivalent to the English verb succeed and refers to achieving a goal. The latter is used to talk about ability but I think the verb saber is closer to English can or be able to.
I'd say 'saber', when it is not meaning 'to know', is talking about skills, which I suppose is synonymous for ability (I did not know how else to explain the difference). But when you say "I can do it myself", I reckon you'll use 'poder'.
I really don't "get" the definition of conseguir in Portuguese. When I look it up in Portuguese - English translations, it's primary definition is "to get." In one source, there were 18 other concepts -- NONE of which were "to be able to" or "can." Given this, I don't understand why "My father does not get to open the gate" is not acceptable. DuoLingo needs to explain this one much better as it seems to be very confusing, especially for Spanish speakers like me who are familiar with the range of meaning for "conseguir in that language. I realize that there may be somewhat of a false cognate thing going on here, but there is significant overlap in meaning in both Spanish and Portuguese and for DuoLingo to continually say that it means "can" when the Portuguese language sources I have googled don't even mention this option makes me think that this may be more regional in usage.