Hi. Long time lurker of these forums first time poster and of course it should be here haha. Anyways I'm a NaruSaku fan so I'll just say my 2 cents.
Irrelevant. It's hard to definitely say what Minato does and doesn't know about the living world other than what he's been told since his revival for one thing.
For another, if it was simply tempers then why didn't Minato make the comparison between Kushina and Karin? Karin was shown to have a quick temper as well and it was even said she was an Uzumaki in front of him but he didn't make the comparison.
And even so, that doesn't explain why he asked Sakura if she was Naruto's girlfriend(and then Naruto answered for her lol). It sort of ties back to Minato's knowledge of Naruto's life because I find it hard to believe he'd see a girl healing Naruto(during a battle no less) and think she might be his girlfriend. My theory is he overheard her speech but eh.
The only thing that shows is Hinata was inspired once again by Naruto. It's nothing new. Notice Naruto hasn't spared a single thought towards the moment since it happened(and even then he had other things on his mind).
So you say but again Naruto hasn't given Hinata any special attention since that moment, called Sakura his sort of girlfriend to his dead father, and now recently her feelings for Sasuke have been put into doubt.
Well there are a few weaknesses in your reply, and I shall point out the reasoning behind it, the first weakness is rather simple, 1. Minato assumed that Sakura was his girlfriend, he saw that Sakura was close by and wondered if that was the case, when you look at Naruto's reply, then compare it with his past replies to the same question, does it really support the NaruSaku pairing? no not really, because his reply again irritated Sakura, the reasoning behind that is due to context, when we look at how Naruto responded when he did to his father it wasn't really a reply that indicated too much romantic interest in the medic nin, but when he replied to the same question to Konahamru, he was more playful, blushing and smiling, again Sakura is repulsed by the idea of being Naruto's girlfriend this is due to the fact that Sakura is in love with Sasuke, there is little questioning it.
weakness number 2. her feelings have been put into doubt, well no, if you look back at the previous page she said she trusted Sasuke, but had a fake smile, Sai noted that what Sakura said was true, but her smile was a lie, now to explain this further Sakura knew that Naruto and Sasuke weren't on the same page, thy were basically at loggerheads with a view between what to do with the Juubi, also the last part of the final paragraph can easily be counter, like all NaruSaku arguments, you have to remember NaruHina moments have come in at certain moment and to force the next one for whatever reason would be inappropriate due to the severity of the situation they are in, so to spare any thought from Naruto's side would be distracting on that part, he needs to do so when everything is finished.
weakness number 3: well this one is one you will not like (plus all of the other NaruSaku fans out there) there is no confirmation that Naruto has gone another level with his crush, sure there has been plenty of times that NaruSaku could have gone one step further, but nothing has ever eventuated, also you say that Naruto hasn't given a single thought after the situation and hasn't given Hinata special attention, well throughout the entire war when Naruto had the interaction with Hinata it has been special, we won't be seeing one for a little while longer, 433 counts as a NaruHina moment because of what was said on the next page.
number 4: Naruto called Sakura his sort of girlfriend, remember you said sort of, again with Naruto's reply, um yeah, I guess if you're splitting hairs I guess... Is that really a reply anyone would have confidence with, now one of my friends is fluent in Japanese he also doesn't pair Naruto with anyone, so he reviewed that small moment and came to a conclusion which I found rather funny. (well it would anger a few people) but that's the thing about being a critical thinker you aren't always going to say things that are popular. my friend said Naruto's reply when he said if you're splitting hairs is Naruto's way of saying that Sakura is a girl and she is a friend, but not in a romantic way. the other thing he noted was Minato was thinking out loud and as I pointed out he jumped to conclusions like 4 other people did in the past and were proven wrong.
on another note, there are two threats to NaruSaku and this one will be a bitter pill to swallow, those threats are not Hinata and Sasuke. those threats are in reality Naruto and Sakura, Sakura because she is still in love with Sasuke and Naruto because he not only accepts the fact that he cannot win Sakura over, but because he has moved on from his crush on Sakura. As I say a fair few times a crush is just a crush, having a crush doesn't mean you will end up with the girl you have a crush on. in the real world do you really end up with your first crush? and I will answer that no you don't (and I speak from personal experience) you are almost assured that you will never end up with you first crush, and I will counter the counter (before you get the chance or anyone else does for that matter) Hinata doesn't have a crush on Naruto, it is a level beyond a crush, she is fully in love with Naruto, and Naruto knows it too.