Many Lan Shang x Shi shippers out there often mistaken Lan Shang as the person closest to romance for Shi. She is not. Unlike Yan Da, Lan Shang was not even a friend to Shi. She was merely a pawn. Let me explain this in detail.
1. Everything that Shi did for Lan Shang was all to protect Ka Suo in the end.
2. (Book version) Shi first saw Lan Shang in her human form (mermaids turn into humans when they reach adult age) when the Ice King introduces her as Ka Suo's fiancée. Shi told his father to give Lan Shang to him, not because of love at first sight, but because he knew that Ka Suo was in love with Li Luo. Since his magic was the most powerful, he told Lan Shang to marry him so he could protect her. Otherwise, if she chose Ka Suo, Shi can kill her whenever he wants.
3. (Drama version) After becoming the Ice King, Shi offered to marry Lan Shang several times but was always turned down. He did not do it out of sympathy or love. He only did it because he was afraid that she will ruin Ka Suo and Li Luo's relationship. Shi did not want Ka Suo to feel guilty or burdened for not taking the throne and marrying Lan Shang.
4. (Drama version) When Lan Shang asked Shi if he loves her, he said he will try to. He was only forcing himself to do so because he wanted her to get away from Ka Suo.
WARNING: HEAVY SPOILERS AHEAD. DON'T READ IF YOU HAVE NOT SEEN KA SUO AND LI LUO'S WEDDING.
5. (Drama version) As I said before, Shi's cold and ruthless personality prevented him from having emotional attachment towards anyone besides Ka Suo and his mom. Therefore, when he felt bad about failing to prevent Lan Shang from getting defiled by Shuo Gang, it was because it would make Ka Suo feel guilty and sad. Same reason applies to her suicide. Some people may argue that he felt sympathy because of morals, but I think it was only due to his respect for women like his mom.
6. (Book version) Did you know how ruthless Shi was in the book? Since Lan Shang was an eyesore, Shi defiled her. This drove her to suicide because having her legs turn back into a tail, meaning that she was defiled before marriage, was a humiliation.
7. (Book version) Li Tian Jin even mercilessly slaughtered Li Jing, who was Lan Shang's reincarnation and Li Luo's lookalike.
**Spoilers ahead** I agree with you in that I don't think that Shi could ever love Lan Shang. I feel like when it comes to love, Shi is a lot like his dad. Shi loves Ka Suo because he was with him when everyone else shunned him. Similarly, Yuan Ji sincerely loves Lian Ji because she saves him when he is at his most vulnerable point. Like Ka Suo, Yan Da has ALWAYS been there for Shi regardless of his status or his behavior towards her. I think Shi does love Yan Da in a way by the end of the story. He respects her enough to call her a friend and if the circumstances had been different (if Yan Da hadn't been torn between doing the right thing and her loyalty to her father, and Shi was able to put himself/Yan Da ahead of his love for Ka Suo) they could have been a great match.
But back to the topic of Lan Shang, I don't think Shi could love her because Shi can't treat her as his equal because he has no reason to trust/respect her more than the average person.
I think Lan Shang and Ying Kong Shi could have like a brother/sister relationship but it's as far as it goes. Ying Kong Shi never really wanted to marry her, he asked her to marry him to remove her from his brother's head. Lan Shang doesn't have feelings for him as well, she's completely blinded by her love for Ka Suo. I think she's a little obssesed, stupid and naive at times. She doesn't even see a difference between Ka Suo and Ying Kong Shi from the back side. When Ying Kong Shi changes into Ka Suo and she knows, it's still okay to hug Shi cause he looks like his brother, she pretends it's him, that's so weird. She lets herself get raped without any suspicion when her eyes are covered and she's touching a guy with armor on, probably desperately wants to be banged by Ka Suo without thinking about anything else. At the end all she can think about is suicide, when she has the mermaid tribe to take care of. When Chao Ya loses her loved ones she's not atempting suicide for example, cause there's still people that she has to take care of as a queen. All Lan Shang can think about is her love for Ka Suo which is annoying and makes her character look weak. What she likes to do in free time is making jewelery and taking care of her beauty. How can Shi relate to her? I'm not sure...In second season he again wants to have some kind of contact with her, probably because she's close to his brother and he wants to know what happens in his life. She is really a good girl and would probably get along with Shi but in 2nd season he reveals he loves fire princess Yan Da anyways, so now it's clear he never felt anything for mermaid princess. And it's no surprise when Yan Da was showering him with love through all the series and even died twice to save him when all he got from Lan Shang was disrespect.