Your batteries are fully charged at 12.8.
source has to deliver a voltage higher than the current battery
voltage in order for your batteries to accept a charge from that source, so that's why input sources are over 13v. If the charging
source were 11v, and your batteries half-discharged at 12.2, they would not accept a charge from that source.
Let's say, by way of analogy, that you have a soft tire on your car. You measure the tire pressure as 15psi. You have an old can of "Fix-a-Flat" that has 15psi left in it. What is going to happen when you stick that can on your valve stem? Nothing. To expect the tire to accept any air, the source must have more pressure than the tire currently has.
So, too, the charging source for a battery
must be of a higher voltage than the battery's current
voltage, or the electrons will not flow into the battery. Because your goal is a fully-charged battery @ 12.8v, the charging source must be higher than that, but not so high as to cause damage to the battery. So, depending on the type of battery, 13v and some change is the right charging voltage.
There are no additional "80 amps" to get into your battery. Battery storage
capacity is measured in amp-hours, not in volts or amps. This seems to be the concept
that is eluding you. Once that is understood, the picture gets clearer
I'll let someone smarter than I describe the details & toss out my crude analogy for something more technically accurate. I write for the uninitiated.