We assign the tasks to ourselves. I meant it more as "this is what I'm going to get done today". If I don't get it done then I'll have to bring it up tomorrow and give an explanation of why, maybe I'm having problems that someone else might have solved but I do not know that, maybe its a large project and I just need more time, I'll still have to mention it. But it gives the other members insight into what, why, and how the task I am working on is going or if someone needs additional help.
And to your next point, we certainly do NOT wait till tomorrow, we reach out in our team chat if we need something and then can discuss further in person.
The stand up is not some strict rigid meeting where its the only time we talk. It is simply an additional small amount of time for the team to discuss whatever, along with what we're working on.
Another key point I did not mention above is the standup helps with those that struggling to speak publicly or just are not familiar with doing so, I have seen noticeable differences from members when we first started to now.
I'm not saying that the stand up is totally relevant and required, I'm sure we would work continue to work just the same without it, but it serves as a small interpersonal connection between members. For my team and I, the stand up works great, for other teams it doesn't.
If you really want to know, try it, or ask your team how they feel about it.