These scripts are often used to install the application, so that introduces a chicken-and-egg situation. How can I install bash before running the installer?

Also, we'd have to build and maintain our own just for platforms missing it, which increases cost and complexity. It's easier, cheaper, and less error-prone just to use a minimum common denominator that runs on everything.

Additionally, when your customers have strict policies regarding how applications are approved for use and have to vet and test everything you give them, it's best practice to avoid installing anything that you can avoid installing. Doing that minimizes the effort and hassle for both the customer and the developer.

How can you install bash without using it? With sh as bootstrapping shell.

Build and maintain your own bash or install script? You don't have to maintain it, it's just building which is already provided by original source build scripts. You can install it as "mysolutionsh", nobody cares as long as you link it to PATH or reference it explicitly. Still easier than avoiding bash everywhere but in personal setups. Yours already familiar with it, that's a waste!