posix-replace-home-string.sh

#!/bin/sh
set -e
set -u

fn_replace_home() {
    my_path="${1}"

    my_delim="$(
        printf '\v'
    )"

    my_path="$(
        echo "${my_path}" |
            sed "s${my_delim}${HOME}${my_delim}\$HOME${my_delim}g" |
            sed "s${my_delim}\${HOME}${my_delim}\$HOME${my_delim}g" |
            sed "s${my_delim}^~${my_delim}\$HOME${my_delim}g"
    )"
    echo "${my_path}"
}

fn_replace_home /usr/bin
fn_replace_home ~/.local/bin
fn_replace_home "${HOME}/.local/xbin"

Z_README.md

Expected Output

/usr/bin
$HOME/.local/bin
$HOME/.local/a/bin
$HOME/.local/b/bin
$HOME/.local/c/bin
$HOME/.local/d/bin