Light aircraft pilot reported an NMAC with another light aircraft in the pattern at BJC airport.

Narrative

Our aircraft was on the GPS 30R approach inside of the FAF when an aircraft was turned base in front of us for 30R. Tower asked if we saw the traffic; I replied yes and that the spacing didn't look like it was going to work out. Tower continued to allow the situation to develop for another 45 second or maybe a minute and then instructed us to go around from approximately 200 feet AGL. We were advised to fly runway heading to the departure end of 30R and then make right traffic for 30R. We executed the missed and offset laterally just enough so that I could maintain visual with the landing aircraft in case they rejected and climbed into us. At the departure end of the runway I initiated a right turn for the pattern and at that moment Local told us to make our turn which would have been approximately a 030 heading. Immediately the Local Controller keyed up and told us to fly north. Then presumably the on-the-job training instructor (OJTI) keyed up and told us to fly southeast. Then immediately followed that up with a traffic alert for [another] aircraft at our 10-11 o'clock and 200 feet high. They have now effectively stacked us on the downwind for the same runway. I wish I could say this is the first or even the worst this facility had handled me; but it isn't. I know they have approximately 11 cpc at this facility and staffing is pitifully short; but staffing aside they do not seem to possess the skill set required to handle the volume and complexity of traffic they receive on a daily basis. I fear something tragic is going to happen at this airport sooner or later; it's gotten that bad.