I think you've all missed Illaoi's characterization. She's not inherently violent, she doesn't live to kill. She simply lives her life to the fullest, following the teachings of Nakaburos.
Her ideology is that you should always live life, keep yourself on the go, always crave and seek out experiences. This is what she lives by. Those who come to her (she is a priestess after all) she tests their spirit, to see if they are worthy of her aid and that of her god. She doesn't just kill people and beat them senseless unless she's given good reason to. At most, for people approaching her or her temple, she would spar with them and test them in that manner, but again, she is a priestess, and death is not her goal; her goal is to open people up to life and experience. Because of her ideals, hedonism plays a large part in her way of looking at life and so she enjoys the company of men like Gangplank, who does live life on the edge, and she wants the company of Braum, who is a celebrated hero and strongman of his home.
Compared to Braum, Illaoi's goals aren't terribly different. Braum wants to make the world a better place - so does she. Braum fights for what he believes in, like her, and he's not afraid to throw himself into the belly of danger in order to protect people, something which Illaoi greatly admires. This is why she thinks they would be a great match. And I think so too.
