I think they should have. As soon as it turned out he knew nothing useful, they should have killed him.
Buffy should have because she's perfectly fine with staking evil massmurderers.
Anya should have, to protect Xander. It seems weird that she didn't.
Xander should have, but both he and Willow can be excused, they were still young and innocent(okay, Willow is just stupid for caring for Spike, after he threatened to rape her in Lover's walk and tried to turn her in season 4). Same with Tara.
Giles is a killer and he kills when he has to. The younger members of the Scoobie gang could be excused, but he's seen too much death to not take Spike seriously.
They let themselves believe he was safe, even tough he mentioned nothing about redemption(not until he feel in love with Buffy anyway), and the chip could stop working any time (or he could kill innocent demons or hold the scoobies while another demon killed them for him). And after him banding together with Adam, they all knew damn well what damage he could do.
I think it's extremely out of character for the entire gang to ignore the danger. So I blame the writers. They wanted to keep Spike around, so they had the scoobies accept him way too easily.
Don't get me wrong, I love his development as a character, but everytime I watch a scene reminding me that they let him live, I grind my teeth.
Btw, my slayage is 666. Mohahaha.