Where a rigid body meets a flexible one, all the bending happens in a few millimetres
Look at the geometry. A 3.5 mm plug carries four conductors in fourteen millimetres: the tip is the left channel, the first ring the right, the second ring earth and the sleeve the microphone — that is the four-segment TRRS plug most phones use. A three-segment TRS plug drops the microphone and puts earth on the sleeve. Either way, the conductors leave the metal body and become four enamelled strands inside one soft jacket.
And that transition is where the trouble is. The plug is a rigid lump and the cable is a soft rope, so every bend in use is concentrated in the ten millimetres where they meet. Copper work-hardens when it is bent repeatedly, then cracks and parts. The PVC jacket, being soft, simply bends along with it and stays perfectly intact. So the fault and the evidence are in different places: the copper is broken, and the outside looks new. That is why "the cable looks fine" is not a test, and why the test has to be an action rather than a look.
Turn the plug slowly while something is playing, then press the cable just behind the plug. Crackling, a channel coming and going, or the sound changing as you press is the answer — the strands are touching and parting at the break. Do it on both sides: a break in the left conductor kills the left channel, and the right one keeps working, which is exactly the "one earbud dead" symptom.
Three faults, three tests
Crackles when you turn or press is the root, and it is the one worth repairing. Cut the plug off leaving thirty millimetres of cable, strip the jacket back fifteen, and burn the enamel off the strands with the iron rather than sanding it — the strands are five hundredths of a millimetre and abrasive takes them with the enamel. Solder to a replacement plug, matching segments to conductors, then put heat-shrink over the joint and fill the last ten millimetres with hot glue: that strain relief is what stops the new joint failing in the same place within a season.
Only works at one depth, or both sides drop out together is the jack or the seating: lint compressed into the socket, or a plug not pushed fully home past the last click. Look into the socket with a torch and a pin — pocket lint packs into the bottom and stops the sleeve making contact, and it is the commonest cause of "it works if I hold it". Clean it out, push it in until it clicks, and test again.
One side only, and the microphone has never worked on this phone is the standard. TRRS plugs come in two wirings — CTIA, which most phones use, and OMTP, which older ones did — and the earth and microphone are swapped between them. Plug one into the other and you get a quiet, hollow, one-sided sound and a microphone that does nothing. Try a cable you know works; if that is fine, the earphones are the wrong standard and no repair will change it.