It's not a bad interpretation
I enjoy poems with a bit of ambiguity and space for the reader. My original intent doesn't matter in that sense.
Don't want to play?
Perhaps you can just help me toward an interpretation?
From a readers perspective: the idea of a river falling is not correct (unless we are discussing gravity and the bending of space). Rivers do not fall in the conventional sense, so the author seems to have taken on a distorted perspective.
The speaker then moves to idealistic if not cliche praise of snowflakes focusing on "purity" and "light." This is juxtaposed with a cynical "'til cold brings its sorrows." The speaker is discussing inevitable change spawned by "cold." Then shortly later describes a glacier, (what could be more cold?) "desperately clinging" to their head.
We see the repetition of "falling" until we see the inevitable result of a "crash." This "'til you crash line is an echo of the "'til the cold.." line. This leaves a feeling of inevitable and impending calamity. The poem offers no resolution for this. There is no after, unless we count "the river tumbles down" as the continuation towards yet another inevitable crash.
The closing line the innocence of youth seems to discuss in part the theme and is offerred as its own sentence as though it modifies the last noun, the river. Rivers are often used to represent life. But this river does not flow or grow, it tumbles, falls, and crashes. This river changes the cold hard way.
The title seems to give us some either literal connection, metaphor or both.
The general tone of the poem seems to discuss an abrupt transition as opposed to gentle one. The speaker tends to have a cynical view rather than one of fondness.
Is that what you intended for your reader?