Personally, I think many bands do have strong accents when they first start out. Oasis are a good example. Listen to the heavy Mancurian accent Liam has when he sings "Shunshinnnnnnnneeeee" on their Definitely Maybe album then fast forward 10 years to the Familiar to Millions live album and he sings "Sunshine" with no accent.
Arctic Monkeys are a new band who sing with a very strong Sheffield accent. But I bet they won't have it in 10 years time either.
Maximo Park, Pet Shop Boys, Nizlopi I could go on...but they've all had/got prominent accents also.
The reason why they suddenly lose their accents? Simple. They all move from where they grew up to London thus their regional accents become dulled but equally they don't pick up a strong London accent either. Therefore when they speak their original accents are still recognisable but aren't strong enough to come through when they sing.
That's just my theory anyway
But then again there are other bands who seem very accentless, Kaiser Chiefs, Rolling Stones, even the Beatles...their accents have never come through like say, The Clash's did. It is strange (and a shame!) I agree.
And I'm contradicting myself now so I'll shut up before I confuse myself even more (!)