You ask:

Why can't I install Sage as a package in an existing Python environment

In fact you can, in many cases.

Sage is a huge Python library with many non-Python dependencies.

Historically, the way to make all these dependencies work together
was to provide a complete distribution.

Tremendous effort has since been put into modularising it
and packaging it for various distributions.

In Arch, Conda, Debian, Fedora, Gentoo, Guix, Nix, and other
distributions, you can install Sage to work with the distribution's Python.

Modularisation efforts are ongoing to make Sage even more modular
and easier to install for an existing Python installation.

This work is tracked at

Welcome to Ask Sage! Thank you for your question.

The question would be easier to answer with an indication of

The usual spelling is Sage rather than SAGE.