Isn't washing a wound in river water a really terrible idea?

No. Running water in rural, pristine areas (which we must assume Gensokyo is) is very clean.

From micro-organisms?

When they have a TVTrope devoted to your topic, consider it suspect.
http://tvtropes.org/pmwiki/pmwiki.php/Main/CoolClearWater

But she's a Tengu.

Things I learned from survival shows: unless it's something like mountain run-off or a heated and sulfurous underground source there's a high chance the water is not clean, running or not.

^What about water that might be used to control a nuclear fusion reactor*?

*Yes, I am aware that nuclear fusion reactors don't work that way... but this is Gensokyo goddammit! Common sense doesn't apply here!