pip3 installs to python3.10 despite the later not being default Python version and shows permission denied

While working on a project, I changed my Ubuntu 20.04 default Python 3 version to Python 3.10, by doing the following changes

1. in bashrc, added alias python3 = python3.10 and sourced it
2. ln -sf python3.10 python3 to mean that "python3 command means python3.10"

I know I messed up and rolled back these by

1. removing the first line from bash and sourcing
2. running the same command in 2 but with 3.8 replacing 3.10

However, now whenever I do any pip or pip3 install, I get this

build-system@buildsystem-H310M-H-2-0:~$ pip3 install pyserial
Collecting pyserial
  Using cached pyserial-3.5-py2.py3-none-any.whl (90 kB)
Installing collected packages: pyserial
Successfully installed pyserial-3.5
ERROR: Exception:
Traceback (most recent call last):
  File "/home/build-system/.local/lib/python3.10/site-packages/pip/_internal/cli/base_command.py", line 160, in exc_logging_wrapper
    status = run_func(*args)
  File "/home/build-system/.local/lib/python3.10/site-packages/pip/_internal/cli/req_command.py", line 247, in wrapper
    return func(self, options, args)
  File "/home/build-system/.local/lib/python3.10/site-packages/pip/_internal/commands/install.py", line 553, in run
    self._handle_target_dir(
  File "/home/build-system/.local/lib/python3.10/site-packages/pip/_internal/commands/install.py", line 563, in _handle_target_dir
    ensure_dir(target_dir)
  File "/home/build-system/.local/lib/python3.10/site-packages/pip/_internal/utils/misc.py", line 105, in ensure_dir
    os.makedirs(path)
  File "/usr/lib/python3.10/os.py", line 215, in makedirs
    makedirs(head, exist_ok=exist_ok)
  File "/usr/lib/python3.10/os.py", line 215, in makedirs
    makedirs(head, exist_ok=exist_ok)
  File "/usr/lib/python3.10/os.py", line 215, in makedirs
    makedirs(head, exist_ok=exist_ok)
  [Previous line repeated 4 more times]
  File "/usr/lib/python3.10/os.py", line 225, in makedirs
    mkdir(name, mode)
PermissionError: [Errno 13] Permission denied: '/Users'

Whenever I put commands to check for the current version such as

1. python3 -V
2. which python

I get 3.8.10. Is this the default for Ubuntu 20.04?

Please tell me why is this happening despite rolling back from 3.10 to 3.8(default) and what do I need to do to fix this?

The main problem here was, that I had somehow put a path starting from /Users in the config.h file whose job was to set the path pointing towards a location where newly installed packages would go. I changed the path to '/usr/lib/Python3.8', which it is supposed to be, since '/Users' was meant for MacOS users, not Debian.

Therefore, this was a problem caused by putting a wrong path.