Yes, there are probably many people who don't realize that an emergency fund is a must-have, usually those who don't realize it are people who haven't felt something urgent. But don't confuse emergency fund and investment, because I also still see people who still equate it, they still think investment can be used as an emergency fund too, yes it can be used but it has risks too. Just imagine when our investment is in a loss, is it worth selling? I think not. I personally always separate the two, because I have to anticipate things that look trivial but have a very big impact.
Unfortunately, there are also people like me who did have something urgent, and that emergency fund was used, and now they have no money in the emergency fund. The worst thing about the emergency fund is that if you have two things in a row, that means you spend it on the first one, and now you have nothing for the second one.
I had something similar just very very recently, let's not put details in because I do not like that, but lets just say, someone I loved had to have a surgery, and I used a bit of my money, and my boss was kind enough to help me too, and that trouble has gone by, it's healed now, we are doing great now, there is no problem there, however after that one another person I care about got sick and this time even bigger issue, and yet I have nearly no money at all, what am I suppose to do when it comes back to back? That's the real trouble.
First of all, I'm sorry for what you've been through recently, when something attacks the people we love, it's a very heavy blow for sure, sometimes if I can choose I personally would rather feel something than the people I love.
Actually from what you experienced, you have done your best, you have completed the first and indeed unexpected things can come even though we have just felt it. Events like this seem to come very quickly.
Although not always, but I as much as possible to divide my income into 2 parts, emergency fund, savings, investment. For me, these three are different things and are very useful for certain situations.
In theory, I would use the emergency fund then the savings fund. This is just an anticipation, if I have to spend it all at once because of something urgent and then the next problem comes, I will probably do everything possible to solve it.