There must be something wrong with your AV-to-HDMI adapter. I use a dirt-cheap box from china, to get my PS2 image and sound (stereo only!) into the HDMI of my TV, and it works like a charm.
I can switch it to 720p or 1080p output. Without analog signal into the RCA-s, it will show a blue screen. So you have immediate visual feedback.
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PS what HDMI-to-RCA cable is it exactly? It can be the inverse cable - one that creates analog (RCA) output from the digital (HDMI) signal. As HDMI is a one-way-interface, it will not work both ways. That is the reason why aliexpress offers both AV2HDMI and HDMI2AV boxes, with input and output sides clearly labeled.
RCA->HDMI involves Analog-to-Digital conversion (ADC), whereas HDMI->RCA uses Digital-to-Analog (DAC) conversion.
Also, HDMI port can be either a signal source or sink, but not both.