Sargent worked his portrait paintings up in layers, as did the majority of painters through the ages. The paintings that he is best known for are not alla prima paintings., in fact his portraits are as far away from alla prima as they could be. His grand portraits, such as Lady Agnew, are far more impressive than the alla prima studies he did of friends and family.
There are a wide range of underpainting techniques and approaches. Some painters do very controlled underpaintings using grays and then carefully glaze over them. Although this method is what most people think of when they think underpainting, it's far from the only approach utilizing underpainting strategies.
Sargent started by laying in color very broadly and refined the shapes and colors on each subsequent layer. Yes, each layer was painted wet into wet, but over over a dry underlayer. Sargent was of documented by his subjects for subjecting them to numerous sittings; in one case as many as 80 sittings. Sargent worked very very hard to make it look like he didn't work hard at all.
Painting alla prima style (in one shot) offers a certain spontanious charm and energy, but just because some artists do alla prima demos (what else can they do in two hours?) doesn't mean they paint this way all the time. There is a big difference between the demos and finished work of the artists you've cited.
Painters like Bouguereau and Gerome started out with carefully delineated inked drawings which they proceeded to wash over with thin loosely placed color. This would be scumbled over and refined as the painting progressed. The very finished look they achieved bore little resemblance to the way things looked at the start.
I teach my students to start with a transparent underpainting, build it up opaquely and further develop it through scumbling. I have added a recent example of this on my website, if you're interested.
http://www.fineartportrait.com/workshop_demo_2.html
When it comes to complex compositions, working without an underpainting is an approach that's dicey at best. Painters throughout history, such as Rembrandt, Van Dyke, Rubins, Lawrence and Raeburn have been well aware of this.
Below is a portrait that Sargent had begun of Edward Wertheimer, who died (unfortunately for him but happily for us) before his painting was completed. It offers us great insight into both Sargent's layering approach and thinking.