I believe the reason why management specifically asked for animated GIFs was compatibility. The GIF format is ancient and supported by everything, makes it trivial to share or embed these animation files.

Ignoring the compatibility, modern video codecs like h264 or h265 are substantially better than animated GIFs, in terms of both size and quality.

BTW I supported them as well. The only target platform of that software is modern versions of Windows. It only took couple pages of straightforward C++ because the OS includes a functional equivalent of libavcodec and libavformat libraries. Moreover, GPU drivers install and register the components to leverage the codecs implemented by hardware of these GPUs. The high-level API to encode compressed video files from a sequence of frames is MFCreateSinkWriterFromURL function.