When it comes to finding out who did what on almost any Rolling Stones album you have to use your ears: Get familiar with the sound of the voices and guitar playing styles of each individual member.
Most people will be aware that Brian never sang a lead vocal with the Stones, but he did sing some backing, most of it can be heard on the first album: “Walking the Dog” may be the best, noticable example: Listen to Brian voice in the chorus, it is raspy with a slight growl, not like Keith’s who had a very smooth voice in the early days.

The next example where you can hear Brian’s backing is “Tell Me”  In this case his singing is much further down in the mix, but you can hear it, along with Keith’s smoother voice. Observe the guitarsound, very similar to “Satisfaction” minus the fuzzbox, perhaps same guitar? Most of Keiths guitars were lost during the stay at Nell Cotes, during the “Exhile on Main Street ” period. The guitars you hear on later albums do sound different, the Strats and Teles became more mainstay instead of the Gibsons.

The last example where I believe you can hear Brian’s backing vocals is “Can I Get a Witness” Again listen to the backing, there are the “Aaaaghs” and the phrase in the chorus. Like before, Brian voice is lower in mix than Keith’s but just listen carefully and you can hear it.

Why do we not hear Brian’s voice on any more songs later on? Maybe due to what happened to the Stones and Brain, it is hard to tell, you need to read between the lines and the facts which are out there.
Personally I think, his voice does have character and it is a pitty that we do not hear it clearer on any more songs, but then again, the Stones changed their music and Brain changed and things became very different.
That first album is very special, great recording, great feel and, even though, many of those first songs were covers, they still sound like they had made them their own with their own feel and style.

Hope to see you soon again,
Eddie