From what I've read, is it correct to assume that the only thing export does is make the variable visible to child processes?
What would be a scenario where you would want to make a variable only visible to the scope it was initialized in, and what would be a scenario in which you would want a variable available to all child scopes?
In general, you only need to export a variable that another process will look for in its environment. How do you know which variables those are? You have to read their documentation.
Whether or not a variable is marked for export makes no difference to the current shell.
Let's construct a demonstration.
$ printf 'echo "foo=$foo"\n' > script
$ bash script
foo=
$ foo=3
$ bash script
foo=
$ export foo
bash script
foo=3
The first time and second time you run script, foo is undefined in its environment because its parent process (the current shell) did not export foo. The third time it is called, the parent adds foo to the script's initial environment because foo was exported.
In response to your comment, the term "environment" has a very precise meaning here. All processes, not just shells, receive an array of strings from its parent on startup, referred to as its environment. There are no particular semantics associated with these strings; it's up to the receiving program how to interpret them.
The shell, for example, ignores strings in its environment that do not have the form name=value, where name is a valid shell identifier. For each such string, the shell defines a shell variable with the given name and value, and marks its export attribute. This is what we mean by an environment variable. You can also "promote" a regular shell variable at any time to an environment variable using the export command, but this doesn't affect the meaning of the variable in the current process.
When any process creates a new process, a copy of its environment is given to the new process. The shell additionally creates name=value strings to pass on from each of its environment variables.
