Semi-baseless guess to hopefully spark others. Are you:

Catch-Up?

I am a food you'd never find or certainly wouldn't eat

Sounds like 'ketchup' but is not

I don't exist, and yet I do, now isn't that a feat!

The concept is intangible

I am a good example of something that is wrong

When you fall behind you must catch up

I'm mentioned in the speech of some but more often heard in song.

Unsure

Clue 3:
Please, submit your best attempt, it might just pass mustard!

I'm thinking 'mustard' was meant to make us think of condiments?

Semi-baseless guess to hopefully spark others. Are you: Catch-Up?

EDIT: New Guess
Are you

Meter

I am a food you'd never find or certainly wouldn't eat

Starts with 'Meat'

I don't exist, and yet I do, now isn't that a feat!

The concept is of meter in poetry and in measurement is intangible. 'Feat' may also be a play on 'Feet' which relates to 'meter' both as a unit of length AND of verse cadence

I am a good example of something that is wrong

The meter of the riddle is wrong! The syllable count is (15-14-13-15), not to mention natural emphases.

I'm mentioned in the speech of some but more often heard in song.

'Meter' can be used for speech patterns (or when talking about distance), but mainly refers to cadence in poems and songs

Clue 2:
I certainly sound food-like, and I mentioned speech and song

'Meter' fits the bill here

Clue 3:
Please, submit your best attempt, it might just pass mustard!

...You put mustard on Meat?