One does not need to weep openly to mourn. To me the Pillar Men just came of as Stoics... Stoicism isn't a lack of caring, it's about self control and not being ruled by emotional impulse and sentimentality.
Kars in the anime did show fondness for his companions in his facial expressions and tone of voice when speaking to them or about them. Also, note that Esidisi wasn't supposed to go after the stone alone, he mentions that the others aren't there yet but that he's so close that he's going to go ahead without them, so the people who interpret Kars as sacrificing Esidisi are wrong, Esidisi took a risk without consulting or waiting for Kars.
In the Anime Kars worries about Esidisi once, in the manga he mentions his concern twice. He says to JoJo that he will not let Esidisi's will be in vein, which shows that his respect for Esidisi's will was paramount.
With Whamu, Kars originally wasn't going to let his battle with JoJo go forward, intending to end Joseph and Lisa Lisa immediately if possible, Lisa Lisa bluffed about a bomb, and Whamu suggested that he be allowed to avenge his master and have his fight. Then Kars allows it.
During the fight, Kars freaks out and screams for Whamu not to use his final form, but it's no use. When Joseph defeated him, Kars did not intervene because this was Whamu's will.
People totally overlook the depth of respect and appreciation Kars showed through his intentions and actions. Originally, he had wanted to rule along side them as gods, not be the lone Ultimate Thing... You'll noticed he lost himself in the final battle with JoJo. He was a bit unhinged, and more than just wanting to avenge them out of pride, he wanted revenge because they were his companions and JoJo took them from him. He wanted JoJo to feel their pain, as in the Hamon scene. That was Kars mourning.
Esidisi is the only Pillarman who had Kars' back, and stuck with him through everything, even raising kids.