Your rewording is correct and preserves the surface meaning of the sentence:
It would be a fine thing if the Muggles found out about us all, and on the very day You-Know-Who seems to have disappeared at last.
As others have noted (but for me not combined clearly) "the Muggles found out about us all" is the "fine thing" the sentence is about. The second part of the sentence is about something good happening, and the first part of the sentence is about something bad happening, so it seems a bit backwards. This is because the phrase, "It would be a fine thing..." here is being used sarcastically. That means it would actually be bad.
So let's reword again:
It would be terrible if the Muggles found out about us all, and on the very day You-Know-Who seems to have disappeared at last.
So it would be bad for the Muggles to find out, but what about the second part that actually is a good thing? A terrible thing happening at the same time a great thing happens is even more tragic because of the great thing happening. Sometimes this is irony, and sometimes it's called irony when it's not really irony. An example of this kind of irony is the song "Ironic" by Alanis Morisette. One of the lines from the song is, "An old man turned 98/He won the lottery and died the next day". Something bad happened to him (he died) right after something good happened to him (he won the lottery). That juxtaposition makes the whole situation more tragic (to most people).
Another example similar to the sentence you are asking about is the common phrase, "Out of the frying pan and into the fire." That saying is sometimes used by people when they escape from or resolve some bad situation and then end up in an even worse situation right away. Getting out of a hot frying pan is good, but not if you fall right into the even hotter fire. One implication is that maybe it would have been better to just stay in the frying pan and deal with that.
So let's reword the sentence again:
It would be a terrible thing if the Muggles found out about us all. And it would be so tragic for that to happen on the very day You-Know-Who seems to have disappeared at last.
So it's really about having something very good happen (Voldemort is gone for good) and then something very bad happening (the Muggles find out). And it's also implied that it would be best if nothing had changed. In other worse, it might be better for Voldemort to still be around and the Muggles to still have no idea that magic is real than for reverse to be true.
This is connected to another idea, which is the belief that when something really good happens, everyone should be especially careful to not let anything bad happen at the same time. This is because it seems like when something really good happens to people, they relax and get careless and might accidentally cause something terrible to happen, which is exactly what McGonagall is concerned about.