Sure, but even now the analogy is not great because the melting part has nothing to do with it. In other words, if you drop an ice cube into a cup of water, it

Sure, but even now the analogy is not great because the melting part has nothing to do with it. In other words, if you drop an ice cube into a cup of water, it will either immediately overflow or it will never overflow. The melting part has nothing to do with whether it will overflow. If the snow on the roof melts -> the cup overflows I think a better analogy is a cup sitting at the drain pipe of a rain gutter and the roof is covered with snow.