After House Baratheon itself nearly became extinct as a result of their own personal follies and vendettas.
House Baratheon, like most of the other houses in Westeros, is a powerful but flawed house. However, what aspects of their house culture would fit into the contemporary world, and what aspects of their behavior would not fit in at all?
Wouldn't: Letting Bad People Gain Power
Although the Lannisters did not Robert's Rebellion until the very last moment, they were keen to prove their loyalty when it was clear Robert was going to win.
And they did so... by murdering almost every living Targaryen in the most brutal way, including the children. The Lannister may have helped Robert in the long run, but the Baratheons never should have let the lions wrest so much power from them when they knew what they were capable of.
Would: Putting Your Money Where Your Mouth Is
If there is one thing that can be said about every member of House Baratheon, it's that they don't say something unless they intend to follow through on it.
They might be terrible leaders, but when Robert chose to overthrow the Mad King he did it, and when Stannis and Renly chose to challenge the Lannisters and each other over who was heir to the Iron Throne, they literally died before surrendering.
Wouldn't: The Desire For War
The Baratheons are historically some of the fiercest warriors in the world, but unfortunately that seems to be their first resort when it comes to solving problems when it really should be their last resort.
While overthrowing Aerys was the right decision, the Baratheons often go to war when an easier and better solution is available to them, simply because their house culture is to fight for whatever it is that they want.
Would: Loyalty
Although the Targaryen dynasty seemed destined for eventual failure, the genuine love and loyalty that formed between Robert, Ned, and Jon Arryn is what enabled that alliance to overthrow the current government and maintain Robert's rule afterwards.
And while the Baratheons can be unkind, it seems that both Stannis and Renly are well known for being loyal to those who offer loyalty in return.
Wouldn't: Primogeniture
Obviously the right of succession is a very important aspect of Westerosi law, but because Renly wanted to be king over Stannis, and because Stannis was very ill equipped to actually be a king, House Baratheon quite literally destroyed itself.
However, through the eyes of a contemporary mind, the answer here seems obvious, and being born first shouldn't give anyone priority over someone who might actually be fit to do the job.
Would: Honesty
If there is one thing that can be said about every character from House Baratheon, it's that they are incredibly up front and straightforward about what it is they want and how they plan to get it.
This has actually historically been a weakness for their house, however it's a somewhat irable trait, especially in a world that seems to be filled with people who are constantly lying and manipulating to get what they want.
Wouldn't: Toxic Masculinity
To put it plainly, House Baratheon is arguably the most macho house in all of the Seven Kingdoms. They are a house of warriors and badasses, and they don't want anyone to forget it.
However, their desire to be the manliest men to ever man often has dire and even fatal consequences for them, and if any one of them could reel in their need to prove their masculinity then literally all of the Baratheon brothers would likely still be alive.
Would: Being Proactive
One of the more irable traits of House Baratheon, despite the fact that it doesn't always yield positive results, is that they are astonishingly willing to be proactive whenever they feel it's necessary.
Many other houses are content to hang back and see what happens, but Robert, Stannis, and Renly have all shown themselves to be very quick on their feet whenever they think the occasion calls for it.
Wouldn't: Homophobia
House Baratheon definitely loves being the tough guys of the Seven Kingdoms, and clearly homosexuality isn't accepted in Westeros at large, but Renly Baratheon feeling forced to stay in the closet for his entire life had a seriously negative effect on him.
It wasn't a very well guarded secret, and it doesn't seem like his brothers treated him badly because of it, but it's hard to imagine that they would have accepted him if it was out in the open either.
Would: Rebellion Against The Status Quo
Granted, things were always guaranteed to pop off once Aerys Targaryen killed Rickard and Brandon Stark, but most of the houses in Westeros were not willing to do what needed to be done in order to depose the Mad King.
It comes as no surprise that Robert would come to the defense of his best friend and the woman he loved, but honestly, not many houses would rebel against the crown like that.