What Tom said regarding capturing the idea, plus...
Why not teach yourself? When it comes to songwriting I am self taught. There are millions of self taught guitarists and pianists who have also taught themselves both theory and notation. Plus there are loads and loads of free resources, so no, I am not excluding a group. Cost prevents nothing as there need be no cost at all other than time. Additionally, to aspire to letting software do it for you ( in the case laid out in replies) doesn't help the songwriter in the long term. Knowledge is the liberator. Skill is the key that unlocks the door. A tool that does that for them makes it accessible in one way but ultimately leaves the writer dependent, not empowered, constrained, not free to express.
Now a tool that augments is different.
I would also point out that the tool I described does NOT write the song for the writer,, it merely suggests the most common next stage in a progression based on theory and common practice. The writer remains filly in control.
If the tool you suggest does not write for the writer either, then surely we are in general agreement? Lol
As an engineer, however, I can happily praise your drive to find a technical solution, while questioning the need for the tool.
Why not sing or play the melody, record it, then use pitch recognition to turn it into notes? Or simply listen to the recording and transcribe as Tom said? Job done
Honestly I am not seeking to ridicule or otherwise, but any engineer worth his salt would be happy to both question a design and question the need for a design... Unless of course you are solving purely for the challenge, but even then you should be happy to analyse both need and design? Perhaps some use case analysis would be helpful.