It's not that they're necessary or not, it's simply the way music works. In order to have the tools I need to give the answer to this question my best shot, I have to start from the beginning, so much of this will be stuff you already know.
Let's assume an equal temperament universe with twelve tones (which is what we have nowadays anyway.)
We know that the pattern
T, T, S, T, T, T, S (where T is Tone and S is semitone) will make us a major scale. Let's start by considering a C major scale.
C, D, E, F, G, A, B, C
That's great. Now, using the notes in this set (which happens to be the key of C) we can write the first bar of Twinkle Twinkle Little Star.
C C G G A A G||
Notice also that these notes are the scale degrees
Tonic Tonic Dominant Dominant Submediant Submediant Dominant
But let's try something different. Let's transpose all of the notes in this scale up half a step. We get:
C#, D#, E# (Or F), F#, G#, A#, B# (Or C), C#
The key of C# major. All of the notes are different. But since we moved each of the notes up by the same distance, the pattern of intervals which we observed earlier still holds (
T, T, S, T, T, T, S). This is a powerful observation. It means that we can use the same scale degrees to write Twinkle Twinkle Little Star as we did in the key of C. In this case,
C# C# G# G# A# A# G#|
That's not that impressive because they're just a half step apart, but try it in G:
G, A, B, C, D, E, F#, G
Here's the set of notes that is in G. Using the pattern of scale degrees that worked previously in the other keys for Twinkle,
G G D D E E D|
Now, so far, this has all probably been stuff that you already know. It's now time to get into the thrust of the question.
A quick aside here is that we've been assuming equal temperament this whole time, but in the older days, different keys sounded totally different, and some were rendered unplayable in some tunings. In that case, it makes sense that we would have different keys -- different keys sounded totally different. (See:
Resuming our place in the equal temperament twelve tone universe which we started, here's a false assumption in your question: "Assume there is no singer involved and we're just talking about what sounds 'good.'"
"What?" you ask. "It's an experimental variable! It's not false, its the premise of the question!"
I think you might be shooting yourself in the foot by excluding that.
Often, what sounds "good" is determined by whether or not there is a singer, and that singer's tessitura, as well as what instruments are playing and keys that they play comfortably in. Other than that, as demonstrated above, the song will be exactly the same, except higher or lower, as you said.
But to play ball with a fingerless world, let's say that we have a magical instrument that is entirely indifferent to keys playing an all instrumental song. Would that song sound better in one key than another? That depends on what you mean by "better." Perhaps you interpret a song require a lower, more thunderous key. Or maybe it sounds better to you played like chirping birds. Either way, you can take advantage of the landscape of the universe of music so that you can play it in exactly the harmonic range you want.
Keys are a natural resource. They exist simply because that's how music works. Skilled musicians can take advantage of them so that they can sing in their range, or create the sensation of chirping birds or a thunder storm.
I hope this is helpful. Good question too!