In my experience - which is being lucky enough to use New Street station about 5 or 6 times per week - lack of platform space doesn't seem to be the main issue. Fair enough I only know what I see and I don't know the full details, but rarely (if ever?) have I seen every single platform occupied (either at the A or B end), though it is an issue when a train at one end is blocking the departure of a train at the other end if they're both travelling in the same direction, and in times of congestion when there's a stack of trains waiting to arrive.
The bigger issue to me seems to be the tunnel bottlenecks. I'm more often sat on a train that is waiting for the signal to clear so it can depart than I am stood waiting for a platform to become available. The volume / frequency of traffic and the fact that there are so few tracks out means a delay to one service can quickly snowball - which then in turn can lead to platform issues as queues begin to form outside the station.
As others have pointed out, the Cross City line, which I also use frequently from New Street, is difficult because of the peak frequency - 12tph combined - and the fact that the line also carries intercity services (which I think join the line at Barnt Green?). That's a whole lot of trains for very little track, adds significantly to the congestion at New Street, and means that barely a day goes by when I don't experience a delay of some degree on the line.
The station rebuild will certainly increase concourse capacity but isn't going to help the above issues much. I don't know enough about the track layouts around New Street to suggest a solution, though I am aware that it's nigh on impossible to expand the current tunnels out of the station. It does seem though that New Street is the victim of poor planning in the past, or at least a failure to anticipate future demand.