In the olden days people used a galvanometer to measure current and voltage. To measure power one would have to read a needle on a dial for two measurements and multiply them together on paper or using a slide rule. To measure total energy delivered from a battery one would have to do that repeatedly and integrate over time using Newton's method by hand calculation.
Being able to take just one reading was a big shortcut, especially for people not trained in engineering, or for whom a rough estimate was good enough. Digital hand calculators, let alone multimeters with built-in microcontroller to do multiplication, did not exist for most of the history of electrical engineering. Considering that lead-acid or nicad battery voltage only varies some 20-30% from full to empty, such may have been an acceptable measurement error in order to allow technicians to take usable measurements without having to do math.
Even battery manufacturers faced the same effort to deliver a W-hr measurement. A constant-resistance load for runtime measurement requires one component: connect a resistor across a battery and measure how long it takes for the battery to drop below the "dead" voltage. For extra credit, plot the voltage over time on a graph. In the 1970's a consumer would have approximated this test by putting new batteries in a flashlight or AM radio and leaving it on to see how long until it stopped working.
A constant-current load requires a few active parts, maybe one opamp and one power transistor, plus a few resistors and capacitors. Measure the time to dead and you have, exactly to the extent conditions are duplicable, a mA-hr rating. Again, measure and plot the voltage over time (ideally at constant temperature) for a more complete picture of battery performance. Such data is sufficient to estimate battery runtime to within maybe 20% under what used to be ordinary conditions: runtime of 5 to 100 hours at room temperature. Just measure current from the battery and divide mA-hr by mA = hours. Good enough for choosing between D cells and AA, quality vs cheap junk alkalines, are your NiCd worn out, etc.
A constant-power load requires multiplication in the load itself (voltage x current in the feedback path), complicating the design and driving up the cost of battery test equipment. Measuring energy delivered requires logging voltage and current, multiplying at each data point, and integrating over time. Possible in a spreadsheet and perhaps trivial with one of today's microcontrollers in the equipment, but a tremendous amount of work 30+ years ago.
In practice, the amount of energy a battery can deliver to a load is highly dependent on the discharge rate anyway, so there would never be one single W-hr rating. If you do enough testing you'll get a chart with power on the x axis and energy on the y axis, where you'd see a moderate to severe downward slope. Additional factors affecting energy storage are temperature, age and usage history of the battery, etc. Further complicating the picture is that modern devices may present an infrequent but heavy pulsed load, which may not "average out" or integrate to the same response from a chemical battery as it would from an ideal voltage source.
Which is not to say that more data from battery manufacturers wouldn't be useful, certainly it would be to those who can make sense of it. Rather that more accurate information would look like a 3D graph, energy vs power and temperature, and that's just the for first charge cycle. The graph could look very different at 100, 300, 1000 charge cycles, under lab conditions where variables are held constant. As most novice engineers don't understand that batteries are not ideal voltage sources, such a tide of data might serve to confuse rather than enlighten most users. In real life, load varies depending on what the multifunction device is doing, temperature varies by tens of degrees, usage varies from minutes to hours per day, on and on. Building a reasonably accurate model of battery behavior (energy storage and state of charge) under such varied circumstances is complex and time-consuming and thus expensive, the domain of university researchers and companies with a business designing battery fuel gauge chips.
In summary, a device does not generally present a constant-power load to the battery irrespective of voltage, and a battery does not deliver a constant amount of energy irrespective of load. Measuring power is (or was) harder than measuring current, and measuring total energy stored/delivered is much harder than measuring rate of energy flow. To a rough estimate current can tell most people what they need to know.