Okay, so with regards to Claire staying. Overwhelming love for Jamie. Like on a fundamental "he's my breath" level.
I also feel like the first book does only a moderate job of showcasing that Claire and Frank were a war marriage. She married him because he was going off to war, and when he has come back and they are at Scotland in the beginning, Gabaldon only does a mild job of portraying the fact that while she does love Frank and finds him kind and smart and generous...she still has a bit of a realization that she didn't know him too well when they married, and he'd been off to war for a few years and expected to just play happy homemakers when he came back and Claire was not about that, especially some of his own ideals about marriage which she disagreed with. The show does a MUCH better job of showing this small, but significant fact in the first few eps. It will become more clear later on, but Claire and Jamie are true loves, and Claire and Frank were a love born of convenience. The more time she spends with Jamie, the more she realizes this. Jamie is a Scot and fiery, independent women are in his wheelhouse...while Frank comes across more as the "husband takes care of the helpless wife". But yeah, as I said, book 1 doesn't exactly nail this fact loud enough....while the show rectifies it.
As to safety, security, and modernity? Starting with the love for Jamie....but also as a healer/wisewoman she feels like she can do some good for the century she's been shunted into...and beyond that she has knowledge of the future and specifically of the battle of Culloden field which she feels she can help with. Is it the brightest decision for her life? Maybe not. But love drives her and the need to help tops it off. She just came out of a disgustingly brutal war (WWII)....and she thinks that if she stays, she can help the Scots. But yeah, the overriding thing is Jamie. Imagine you met the absolute love of your life. The epitome equal of you. Would you leave them for conveniences and someone who you didn't love this fiercely?
Jamie is an 18th century Scot. As far as all the males who surround him, he is actually a saint....in comparison to say Dougal....who does what he wants with who he wants and is still called a hero. It was the age he was in. So you can't blame him for his actions since this was how he and every other male on that continent was raised. Also, trust Claire...she's not one to stand down when she feels slighted.

(Sidebar: And this is nothing to what Black Jack Randall thinks appropriate! LOL)
I immediately liked Jamie more than Claire to begin with. Though flawed by his era, he is a stalwart, heroic, and genuinely easy-going person to root for. Claire is a slow boil. By the second half of book 1 I started to see her in a more interesting light. She makes HUGE, sweeping mistakes and is super flawed in her own right...which made her more interesting as a character. But yeah, she grows on you much slower than Jamie does. Also, Claire has a built-in irrationality that I find compelling.