I agree with others here that appearance is regrettably important in general.
You ask about looking good for a romantic partner. I suggest you should additionally think about screening effects. Remember that, as a generalisation, the typical woman faces a different problem than the typical man when looking for a partner: the woman has to screen through a large number of offers, many/most of which are low-quality, to find a partner she wants. People in general respond to screening problems with shortcuts and heuristics. A person who knows you well will consider your appearance as one aspect of what they know about you; a person who has just met you is likely to make a snap judgement on your appearance. Your aim should be for an appearance good enough that it is not making most potential partners screen you out at the first step.
As several commenters here have said, the business owner example isn’t a great fit for heroic responsibility. The core is taking responsibility for things that aren’t your job, that you are not socially expected to be responsible for, because you have decided that the thing needs to be done.
The archetypal fictional example is the hero who raises the rebellion that overthrows the Evil Empire. A normal sensible peasant whose home has just been burned doesn’t do that, he just tries to survive the winter. The hero decides to do more than that, even though it’s not their job and they’re not (at the start of the book/movie/whatever) distinguished from all the other thousands of people in the same position. That choice is why we call them the hero.
Or, for a real-life example, a factory-owner in 1940s Germany might reasonably feel that he has quite enough to do trying to keep his business running in a war zone. It’s not his job to save Jewish lives, they’re not his relatives or anything. And even trying to save them could be very hazardous. Nevertheless, Oskar Schindler decided he was going save as many lives as he could anyway, and saved about 1,200 people. That’s what heroic responsibility looks like.
And, the most obvious example to anyone on LessWrong: there is no social expectation that a normal citizen who is not a senior government official will decide that it’s up to him to save the world from the risk that Unfriendly AI kills us all, but Eliezer Yudkowsky saw that no one else was going to do the job, and he wants the world to be saved, so he’s spent his career trying.
The heroic responsibility described by OP, the determination to focus on whether [goal] is actually happening or whether people are buck-passing or symbolically pretending to do the thing, or whatever other failure mode—all that is downstream of the heroic decision to take responsibility for [goal] in the first place.