[sciurius]

Just a couple of remarks.

"A/Db" is blatantly wrong.

If we are considering an A chord with its major third in the bass (aka. 1st inversion) out of context I tend to agree that "A/C#" looks more natural.
However, when it is part of a chord progression in a piece written in the key of Ab or Db, I think "A/Db" looks more natural.

"The key of F uses flats" is a misconception and not true at all. The key signature has one flat, yes, but it does not mean that there should only be flats in the song.

No doubt you have noticed that there are no chords with sharps in the Yesterday example. Browsing through my Real Books I have never seen a sharp in a chord in a song that is in a flat key.

Having non-flats in the song itself is possible. IIRC, the Yesterday example shows something called a transient key change in the 2nd measure.

In songs in D minor - with the same key signature of one flat - it is common to have A major or A7 chords with C# notes

Of course you can have "chords with sharps" in a song with a flat key, but that is not related to the issue at hand.

Summarizing, I'd opt for writing the chords and bass according to the key of the song.