There are already places where precision cookers are already used for "flavour layering". Just take a look at "step mashing" (also known as "step-infusion" mashing) used in brewing some beer styles.
With step mashing, the grain and water are first heated to around 50–54 °C for enzymes to break down proteins. Followed by one or two heatings to 62–67 °C and 71–72 °C to convert starches to sugars. There was a recent Anova blog post on how you can use the APC for mashing with the Brew-in-the-Bag method of brewing. (I wouldn't use this method for something like gumbo but the concepts are the same.)
Many slow cooker recipes have similar one-pot flavouring dishes such as starting at high for a few hours and then stepping down to low for a long cook. Just that most slow cookers are not as precise as most sous vide cookers and ovens.
I imagine one can make jambalaya by using good resealable bags, adding additional ingredients over the cooking period.
As others have mentioned sous vide is another tool where I would often tell students "Use the best tool for the job at hand."