Corporate aircraft on visual approach base to CHA observed aircraft lights; questioned ATC; and absent a timely response; initiated a go-around; reporter claiming recent CHA ATC services were questionable.

Narrative

Approach cleared us for a visual to Runway 20. Prior to turning final and requesting a frequency change to Tower; pilot flying asked what the lights were. Realizing it was another aircraft; we started a right turn out and go around at the same time TCAS alerted us to traffic. Tried calling approach and got no answer from them on 2 calls. They answered on the 3 call. Tower had cleared the air carrier for take-off on Runway 2 and Approach cleared us for 20. They told us they were watching us when we questioned what was going on. After landing; asked Ground Control if we needed to call and they acted as if they didn't know what we were asking about. Chattanooga TRACON is an accident waiting to happen. These controllers have gotten worse and worse over the past few years. They can't do position and hold now because they say they are understaffed. They need someone watching the local controllers. I have been flying in and out of this airport for almost 30 years; and the past few are getting crazy. This is not the first time this type of situation has happened with my company.