Your hand-drawn schematic clarifies a great deal. To produce voltage to the battery the triac fires, conducting AC through the transformer. Probably the IC detects voltage, and automatically varies the timing of the triac.
Since you have 5V DC, then the circuit near the TL431 must be working okay. Yes
There are several components where a fault can prevent operation. Look for bulges on electrolytic capacitors. Test them if possible. (Some technicians. simply replace them as routine.) I checked the ESR & all were OK. There are no bulges. I'll remove them from the PCB & measure their capacitance
One or more experts here spoke of optoisolators frequently going bad over the years. Yes, I intend to check the Opto as it is in the feedback path.
If you apply low voltage AC to the transformer primary, you should read much lesser voltage among the diode bridge connected to the secondary. (I would not advise connecting house voltage directly to the transformer.) If just one of the 4 diodes goes faulty (open or short), it can cause further faults. I checked the diodes & all are OK.
You can troubleshoot by bypassing certain components, or by providing a substitute signal at certain places. For useful diagnosis you need to attach a battery, which is risky. (You can try attaching a resistive load however the smart charger might decide battery voltage is non-existent therefore it shuts down operation.) I've already checked the transistors, diodes & resistors. I may need to remove the capacitors from the PCB to measure them. I don't think the charger would be fooled by a resistive load. If the microprocessor is faulty I won't be able to replace it as I would need a copy of the software in order to use say a PIC.
All your tests may lead to a conclusion the IC really is bad. If you can't fix the charger you might decide it's worthwhile for you to modify the existing circuit and install a simpler charging circuit. It depends on how you gauge your abilities, versus the risks to battery wellbeing. I don't know what signals come from the NTC & COD lines. So it would be very difficult to design another charger. My abilities are not an issue. I have a degree in electronic engineering & much experience with designing & constructing.