What are the advantages of DSB AM with carrier?
Simplicity of modulation and demodulation is indeed the chief advantage of DSB-AM -- a DSB-AM signal can use a low level modulator and Class C amps or a high-level final stage modulator with Class C carrier drive, whereas a SSB or DSB-SC signal must be generated low-level with a doubly balanced modulator and amplified linearly, which requires more components and reduces amplifier efficiency.

On the demodulation end, SSB and DSB-SC require either an accurate BFO/carrier inject oscillator or a synchronous demodulator such as a Costas loop, both of which require many more components and are much more expensive than the triviality of a diode demodulator, which only requires a tuned circuit, a sensitive (point-contact "catwhisker" detectors and germanium diodes were used in the bad old days; nowadays, a fast Schottky is the diode of choice) diode, a RC network, and a sensitive earpiece or audio amp.