I'll replace the USB port as soon as it arrives. It is a hassle and also risky, but worth a try on my SGS3 that won't charge in ways described earlier in the thread.
The list of possible causes:
1) weak/partially broken cable
2) weak/partially broken plug
3) broken charger, or charger too weak
4) broken USB port
5) electronic problem on the board (current negotiation with USB peer device)
6) software or firmware problem for energy management (possible combination with 5))
However, none of the explanations really fit to full satisfaction, mostly because of the gradual degradation of the phone's ability to accept a power source.
A weak contact remains weak even if you exchange the power source (charger vs. USB port, different current capabilities of power source), and a broken USB port should be somewhat receptive to the angle and pressure with which you plug in the cable. So - if the cause for the failure is really physical/electronic, then it seems logical that there is more than one cause (in light of the inconclusive effects as observed). Considering combinations of the causes above, 5) and 6) appear to be on board all or most of the time.
One more:
Samsung Galaxy S3 GT-I9300 Charging Problem Repair solution - mirrorfy says that there are two capacitors (are they capacitors?) to exchange. I'd say that this is an even tougher repair project than the USB port exchange; soldering at distances that small can very easily cause overheat damages on neighboring parts.
John, you have mentioned the "charge log file" on the phone. If a history-keeping logic keeps track of charging/discharging cycles, it runs full earlier with more cycles per time (such as for Ingress players...

), and may become susceptible to failures with more data. eg, it could fail due to a timeout of an asynchronous operation/interaction between CPU/GPU power management and charging management, causing something like a deadlock, in a more true sense than usual. Example: If the phone starts up from a complete power loss, the CPUs are initialized and run at high speed until they are shut off or slowed down. In a power saving mode, the phone's performance is decreased for the benefit of longer battery life. If there is a condition at which the phone needs to evaluate a mass of data before it can make a decision, that operation may take considerably longer if this happens with reduced performance, thus leading to a temporal conflict between electronics and logics. In addition to that, the drain of energy from making decisions about accepting power sources (determined by negotiation on the USB port) may lead to a loop from a 0-powered startup and a shutdown to 0-power due to insufficient energy. Which is what I observe.
Again, to make this clear: This is about ideas and is "informed speculative", in lack of anything better.
John: Where can I find more information about the charge log file? Where would I find it?
About Samsung:
I understand that Samsung is hesitant to admitting that there is a problem in an entire family of phone products: It is expensive to handle as a company, and it causes bad press. The damage is certainly substantial.
However, since we definitely SEE a systematic failure with the phone ("systematic" as in design and construction), Samsung could actually benefit from the community's contributions and offer solutions and information, no matter how difficult, expensive or ugly it is. At least this is a possible and most likely effective measure against a legal battle that Samsung is unlikely to win. In Germany, there is a legal obligation for a 2 year warranty.
Thanks,
Roman.