When a package's exact name is not known (python3, python38, something scl) file name searches can do a more complete search: yum provides '*bin/python3*'

Correct, there is no python3.8 in EPEL 7. There is a python3.6 in EL 7, and EPEL policy is not to replace EL packages. Plus the distro's policy to avoid major upgrades, would seem like it is stuck on older Python.

Your yum repo list hints that CentOS software collections are enabled, which could be an alternative.

yum install centos-release-scl-rh
yum install rh-python38-python

Note both the package name and the files are prefixed, which allows coexistence with other pythons. While perhaps SCL is not the most popular packaging convention, it is already built and public, an advantage over your own source builds.

Currently CentOS 7 is 87 weeks from end of life, which is not a lot by EL time scales. Most of the maintainer attention is on EL 8 and 9, 7 is critical fixes only.

Budget some of that time to figure out what you are are going to do post CentOS 7. Which may be non-trivial, as CentOS 8 is dead, and if Stream is not what you need, may be switching distros.

By default, CPython install compiled python binary in /usr/local/bin/python. (source code: https://github.com/python/cpython/blob/main/configure#L571) You may specify the prefix mannually as configure --prefix=/usr.