No, they don't. The UnitedHealth dude who got shot had a CEO title, but Thompson was ultimately a middle manager.
The actual CEO of UnitedHealth Group–the one who signs off on its financial statements and fields quarterly calls–and the billionaire owners were fine. Which explains, in part, why nothing changed after the shooting.
It would have sent the message its sender (and his supporters) intended to send. Instead, both sides got a convenient totem. Luigi's supporters get to pretend he was effective. The billionaire class got to consolidate power here and there by pretending he was more than a one off, and by pretending he was competent.
One getting shot is an anomaly; if one were being shot every week, then the system they operate under would be different.
(To be clear: I am not advocating for changing the system in this way. A world ruled by vigilante justice is not a good world to live in.)
Health insurance companies don't have absurdly high margins. They offer more expensive plans and less expensive plans. The less expensive plans will cause more claims to be denied (not covered by the cheaper policy), the more expensive plans cost more.
People both can't afford the more expensive plan and can't afford to have a claim denied and then get upset, but the company can't actually fix that because it's caused by the regulatory environment. The only way for them to deny non-trivially fewer claims is to charge higher premiums. To actually fix it you have to solve the scarcity problems -- get rid of certificate of need laws, open more medical residency slots, require price transparency, etc., so that people can afford the plan that denies fewer claims. But those are all things the government would have to do rather than the company.