^ Well, nudging does make things less rigid. Listen to dance tunes that have a certain motif that you can't count until the beat comes in.
Jazz/Latin is very popular for having off-beat rhythms and I don't mean just the drums.
Sambas usually have syncopated bass lines (by anticipation) - notes don't line up directly on the beat. Meanwhile, the harmony slides of the pulse/beat as well.
To the ear, it doesn't sound like it because everybody is playing together when it's all in context.
When I write harmony using strings, the harmonic rhythm is most often a minim (half-note, two beats); yet, they are often a quaver (eighth) earlier than the pulse. In context, one may not be able to tell.
as far as making adjustments, that depends on how rigid you want it - at the same time, you don't want a take (or ten) where the player is off every time or gets worse with each take.
Then, if it's so be played by someone else, it's obvious one may not want the performers to get really crazy with it. Not as rigid as the traditional orchestra, but they mess up sometimes, too.
Always allow for imperfections, I guess is what I'm saying. In this case, it's not a feel thing as much as it is that imperfections are inevitable.