Can transposing my song help me understand what's wrong with it?
If you've written it in the wrong key to begin with then, yes absolutely.
More to the point, it helps if you have something like a "composer buddy", another pair of ears to listen to what you've produced and give feedback. And, yes, probably worth trying different keys as well.
The key of composition can make an enormous difference, particularly on some instruments.
Do you choose your original key based on what you are trying to achieve? If so and you don't like the finished product then maybe you failed to achieve what you wanted or, more positively, you achieved something completely different. In that case maybe you do need to reconsider the key?
If you don't make a conscious key decision based on mood, etc., but just on what's easy then perhaps that is your problem.
To give an example of the difference a key change can make take the example of Dvorak's Humoresque No7 in G♭ Major. Originally written for piano it is also very popular for violin. Top violinists play it in the original key and it can sound hauntingly beautiful. However G♭ major is difficult for lesser performers and at amateur levels it is normal to play it simply transcribed into G major. The problem is that on the violin this is a more unambiguously happy key. Everything rings. G♭, with its 6 flats including all the open strings, is more moody. Nothing rings.
When writing, recording, or making music on the computer
It also helps a lot if you can play a real, physical instrument, even if it's just a piano. If you limit yourself to just the computer then you miss out on a world of possibility.