Loudspeaker volume issue
I finally stumbled upon the issue. It is the wires. I used wires from a generic ribbonstrip of 40 wires (male-to-female, Dupont type). I would cut off one end and solder them on to the speaker and use the other end to connect to the amp. For some unknown serendipitous reason, I measured the resistance of a wire, from speaker to amp with my multimeter. The resistance was 80 ohms. The other wire was also roughly the same. So, I was running the sound for a 4 ohm speaker through an additional resistance of 160 ohms. Replacing these with solid core wires (of ~1 ohm for similar length, ~10") got the volume to acceptable levels.

Of course, why in the world would a wire of 10" have a resistance of 80 ohms? I don't know. But this was a consistent measurement across several samples from that strip. 

@glen_geek: But the motorboating did not end: After certain input volume increase the effect is recurring, though at a much higher volume. I remembered your suggestion about power supply weakness being a cause. I checked and again, I was using the same wires. I removed the wires from the amp and directly connected the amp to the power supply rail pins. Motorboating solved. Increase is now smooth all the way to maximum on the input signal. It is really wonderful to hear the same speaker sound so clear and loud, all because of a change of wires!  

My thanks to you and jonk.

tldr: Check your wires. Power wires and audio wires should be low resistance. (&lt;1 ohm per foot?)