Have you stopped clubbing baby seals? Just a "Yes" or "No".
You pose an invalid question as neither amp-hours nor watt-hours does "all of those tasks". They are only part of a total package. To be accurate, either method requires supplementation. As I said earlier, for power loads we use watt-hour calculations and for current loads we use amp-hour calculations.
Now if you want to be picky, and you have a system measuring amp-hours and I have a system measuring watt-hours, I can make a battery management system that is much more accurate and versatile than you could ever hope to achieve. It is only logical because I would have more field data than you (the voltage in addition to the current and time). The point remains that neither watt-hour counting nor amp-hour counting alone make a good battery monitor because that is not their purpose. But either method can support a good battery management system.
If what you have is "watts", not "amps and volts = watts", and what I have is "amps", my system will be more accurate than yours.
The half-cell reactions for batteries do not involve "watts", they involve charge. Each reaction results in 2 electrons moving from one half-cell to the other. Again, no "volts", just charge.
Pb + HSO4- -> PbSO4 + H+ + 2e- <== negative plate reaction
PbO2 + 3H+ + HSO4- + 2e- -> PbSO4 + 2H2O <== positive plate reaction
An amp is a coulomb-per-second, and a coulomb is 6.241x10^18 electrons. Multiplying amps by time gets you coulombs, which gets you reactions times two. From the quantity of reactants, you can get the total number of coulombs that are available at various concentrations (which are based on state of charge) and temperatures (since thermal energy is required for the reaction to occur). And there is are still no "volts" in that set of values.
What you need to know, to determine the remaining charge, is the stoichiometry of the battery at any given instant in time, plus a bit of Physics. That is, the quantities of reactants in the two half-cells and the temperature in which the reaction will take place, activation energy at given temperatures, etc. State of charge is the integral of "reactions so far", so you can derive reactants from that. And that is all. Volts aren't even in that equation.
If you'd like to present some Chemistry or Physics which disagrees with that, I'd be more than happy to entertain whatever science you've got to back your arguments up.