I can see why things might be keen as mustard because mustard does have a pretty sharp taste which does bite the tongue rather like a keen knife.
But cutting the mustard?
There are plenty of ideas about how this expression came into being, but none of them really, well, cuts the mustard as far as I'm concerned.
There are those who say that mustard is a stringy crop and if your tools aren't the best then you won't be able to harvest it; there are those who say it's a variant of cut the muster (the problems here being that a) the phrase cut the muster doesn't seem to have existed before cut the mustard, and b) cut the muster doesn't actually seem to mean anything).
The cut in cutting the mustard may be the same sort of cutting that means showing, as when someone cuts a fine figure. Mustard has been a metaphor for excellence since at least 1672.
As if that's not baffling enough, then there's the mustard seed that's used as a symbol of smallness in the Bible:
He [Jesus] said, “How will we liken the Kingdom of God? Or with what parable will we illustrate it? It’s like a grain of mustard seed, which, when it is sown in the earth, though it is less than all the seeds that are on the earth, yet when it is sown, grows up, and becomes greater than all the herbs, and puts out great branches, so that the birds of the sky can lodge under its shadow.
Mark 4 30-32
Jesus would have been talking about a black mustard seed. It's nothing like the smallest seed on the earth* but it was probably the smallest seed Palestinian farmers were sowing at the time, and perhaps then the word seed then implied something which is sown rather than something from which a plant grows.
Sorry, that's rather a lot of don't-knows.
Still, the derivation's interesting:
Word To Use Today: mustard. This word comes from the Old French moutarde, from the Latin mustum, which means the newly pressed juice of grapes ready for fermentation. Originally, this was what was used to mix with the crushed seeds to make the mustard paste which none of us really likes very much.
*That would be the seed of some tropical tree-growing orchids. The seeds of these can be too small to see, and you might get as many as thirty five million to the ounce, or one and a quarter million per gram.