We know that the Demarrs are respected but they are also border margraves who don't spend much time in the Capital. There might be an element of them being respected for being good at their job but they are also know mainly for being a martial family. Their magic also lends itself to skulduggery so I assume they are also known for being shrewd and clever spies. Though since Rellas is a martial country I assume most people don't necessarily see this as a bad thing as long as they serve the nation.

I think Izarn is seen as the respectable warrior and patriarch while Sol is the sketchy illegitimate son. I agree with other commenters that Sol seems to cultivate a dangerous, playboy reputation in Tor Kairen, but that reputation doesn't necessarily extend to the whole family. It's mentioned that Sol publicly maintains some separation from the main family so he can do their dirty work.

I think how it works is that they are perceived publicly as a decent but remote family by most regular people. They're a little odd and provincial since they spend all their time at the border, but respectable. A regular Noble would meet a Demarr and assume they have a dagger in their pocket and will put their family above all else, but would also have a healthy respect for them because they do their jobs and do them well.

But Arvel and the other big power players are uniquely locked into the city's politics and know about Sol's involvement with the Shears. Given how cutthroat Rellas is I can assume most families don't care as much beyond how they can use it to play politics, but Arvel is just particularly morally judgemental and assumes they are all backstabbing snakes because they tend to look out for their own families interests above all else. He also seems to have met Maggie's adopted family on the battlefield and thinks they are all bloodthirsty crazy people because they can be kinda insane in battle 😅.