For an example if I used an escape tone from F major to C major, using a B-flat note to G, once the chord changes from F major to C major. In all examples I've seen on non-harmonic tones it seems you have to start from the chord tones itself, then step or leap up to the non tone, but can you possibly just play the non chord itself without having to approach it by any of the chord tones?
Of course you can! Sometimes (but not always/everywhere!) these nonharmonic tones can be called "incomplete." Imagine you have a C major chord; if the melody plays
F is a neighbor tone. But if the melody plays
C--F--E, the F is suddenly an "incomplete" neighbor because it wasn't preceded by E. The same would be true if you just had
F--E, either with or without a rest at the beginning.
You can even take this to higher levels. Beethoven's first symphony, although in C major, starts off on a C7 chord, actually the V7/IV in the overall key. And then you can get to Chopin ballades, etc., which actually just plain start off in completely different keys than what the piece is actually "in."
All of this to say: do it!
Yes, you can start anywhere you like.
We learn by analysing simple music, where everything is neatly in-scale, in-key, in-mode. That's fine But don't fall in to the trap of thinking you're learning a set of rules, a list of notes that CAN'T be used.
Anyway, 'theory' knows about 'unprepared suspensions'. Sometimes called 'appoggiaturas'. You can even have suspensions - prepared or unprepared - that don't resolve at all!
'Theory' is much better at describing and labelling what you HAVE done than at telling you what you CAN do.