You nailed it. There is a universal way things are most often sung, just because it sounds better. The way things are pronounced also may vary according to whether it’s a single singer, or a group giving a choral performance. In choir, I was taught how to say things so they came out sounding “correct” to the audience. For example, our instructor preferrred us to say “Al-LEH-lu-yah” not “Al-LAY-lu-yah” because if everyone sang it the first way, the audience would hear it the second. I was skeptical (though I sung it the way he wanted) until the day he recorded us in class and played it back for us.
First person i thought of when reading the OP was Billy Bragg. Listen to him and tell me there’s no accent. 
Also, many Australian singers have quite distinctive Aussie accents, even when they’re singing. Peter Garrett, lead singer of Midnight Oil comes to mind, as does Craig Bloxom of Spy V Spy and Bernard Fanning of Powderfinger.
But i think it’s certainly ttrue that singing tends to smooth out many people’s accents. The simple mechanics of singing seem to do this sometimes.
Same way with Rs. If a choral group sings “forever” it sounds like “foreverrrrrrr.” If they sing “foreveh,” it sounds like “forever.”
Beatles used native accent only on “Polythene Pam” and “Maggie Mae.” “Penny Lane” also sounds more English than most Beatles songs, but English lite. A lot of their singing, “Eleanor Rigby” is a good example, is in the compromise accent called “Mid-Atlantic” (i.e. halfway between British and American).
Singing seems to soften an accent but sometimes it’s still noticeable. Cerys Matthews from Catatonia sounds incredibly strongy Welsh when she sings.
However a Welsh friend of mine used to find her singing irritating, claiming she was deliberately “putting it on” and exaggerating her accent.
The Proclaimers (Scottish) strike me as another band who sing strongly in their national accent.