And I think it was easier for Kevin to admit he needs professional help because he sees himself as a mess, and he believes people expect him to be a mess. So he has no perfection illusion about himself, and no false believe that he is strong enough to over-come his "weakness". People can't think less of you for getting help when they already think the worst of you.
Randall on the other hand, spent his whole life believing that can be perfect if he just work hard at it. That he has the ability to be perfect, therefore, if he isn't, then it is because he is not trying hard enough/making the right decisions. This idea is reinforce/praise over and over again in different ways and levels by his parents/siblings/wife/others. To Randall, people telling him he needs help probably feels like people telling him he is failing/failure, therefore, the solution is not to listen and get help but to work harder at being perfect. What people say to you when you are mentally ill and what you hear them say are too very different things. So in order for someone like Randall to accept and knowledge he needs help, he needs to be knock down so hard that he can't get up, or think his way out off the hole he falls into. As long as he is functional and able to come up with reasonable ways to "fix" himself, he will continue to believe he can do it on his own.
This is why I like to judge Kevin/Randall/Kate differently from each other because their weakness/strength are different, therefore, they are going to react to the same thing differently.
Also, like to add that mental health as a serious medically issue and not a moral/personal failure is recent development that majority of people still don't really accept in themselves and others, even in the medical field where people should know and know better.