How can we run the program generated by STM32CubeMX without having to use the GUI?

[TDK]

You need to compile it with an IDE such as STM32CubeIDE, then upload it to the chip using the IDE or STM32CubeProgrammer.

You also need to add your own code to make the program do something useful.

[Mmohs.1]

What file(s) need to be uploaded to the chip? We used STM32CubeIDE to build the application, but we want a simple way to load & run the application without having to start up the IDE and click on buttons.

[TDK]

The *.ELF file needs uploaded which is in the build directory (typically Release or Debug). Sometimes the ELF is converted to a HEX or BIN which, for this purpose, are equivalent.
It can also be uploaded with the STM32CubeProgrammer command line interface.