Why is CDMA so complicated, wouldn't this simpler solution work?
Your solution works, and it is known by the name "time division multiple access" (TDMA). Basically, you assign time slots to each transmitter.

CDMA has several advantages compared to TDMA, but it also has its own disadvantages. Both schemes are inferior to more modern techniques such as OFDMA.

Main disadvantages of TDMA:


With data rate constant, the time slices become narrower as the number of transmitters increases. This means that the bandwidth also increases. With CDMA, you can increase the number of transmitters without increasing the bandwidth, by adding orthogonal code words.
TDMA requires very precise synchronization between transmitters. It is very hard to keep a large number of mobile units synchronized so that they won't transmit during the time slot of another unit. This may be solved using guardbands, but that decreases efficiency.


A couple of advantages of CDMA:


More resistance to fading and multipath. CDMA reflections are orthogonal to each other, so they cancel out.
If the codes are private, then there's a measure of inherent privacy.