We know that, due to his tutoring (likely by Sulyvahn or Aldia), Lothric believed that linking the flame and prolonging the cycle was ultimately pointless, so he refused the sacrifice
Lorian was a badass knight captain in his prime who protected his brother and slew the Demon Prince singlehandedly, but when we find him, he's in a pretty sorry state
The previous Lords of Cinder were resurrected at the beginning of DS3, but Lothric never became a Lord.
It's stated that Lorian "took on his brother's curse and was left mute and crippled by it"
We're told that the twin brothers share a soul
I would hazard a guess that Lorian tried to take over his brother's duty/"curse" and sacrificed himself to the flame, so his brother wouldn't have to; however, because the two brothers' souls are shared, his soul alone was not enough. The linking failed, and flames consumed him. He was left a crippled husk, and it seems like only Lothric's magic is keeping him going at this point. During the fight, if you kill Lorian, Lothric resurrects him, but if you kill Lothric, Lorian dies immediately regardless of how much health he has.
Who else do we know that's linked the flame before and was crippled by it? Ludleth of Courland.
Ludleth is missing both his legs and is tiny and frail; however, we know in the past that he successfully linked the flame, so he was presumably pretty strong at some point. Again, we know that you can't successfully link the flame unless you have a very strong soul. Lothric still qualified because he had the lord's blood in him (Gwynevere very likely being the Queen), but Ludleth was a human king, so he may have been a great warrior like Vendrick. He refers to his own body as a "charred corpse" and a "sad cadaver", so this clearly isn't how he looked before.