Merely being in the UK is insufficient. You have got to be exercising Treaty Rights, and not just as a visitor.
One way of doing that is being a person of independent means, but the lack of bank account, and the need to live on the streets at times clearly shows that you were not in that category.
Could I just send them my story along with my Belgian passport and do the citizenship test and then apply for naturalisation?
Well no one will stop you posting the application, but on the facts as outlined here you have not held Permanent Residence in the UK for at least one year. Indeed it is extremely problematic you proving that you acquired such status automatically on 30.04.06 when the EU regulations changed. And you need to have held Permanent Residence for at least one year before applying anyway, so any application posted before 30.04.07 is bound to fail, given the lack of Permanent Residence sticker in your Belgian passport.
In short, this is not going to work. If you want to get British Citizenship you should move back to the UK and start exercising Treaty Rights, such as by being employed. After 6 years of doing that you might qualify to apply for Naturalisation.