100% correct - Mental health issues have nothing at all to do with weakness.
As I posted last month: in the last year I have battled (& hopefully beaten) bowel & liver cancer, coming to terms with the result of the treatment (permanent colostomy bag) and the deep depression & insomnia it all...
You are so right, we should all look out for the signs and and offer support and listen if someone in need approaches us.
I'm really sorry for your loss but echo the OPs - it is not your fault and you should not beat yourself up. And yes definitely talk to those close to you about how you are...