You are not required to use Composer in any particular way. I recommend using it in the way that works best for you. That will depend on the type of development and release process you use. It sounds like you are using a fairly short and simple process where you develop for a while, decide the code is ready, and upload it to your production server. With this process, you could simply upload everything, vendor directory and all - just as you state. This could lead to unused files/directories not being removed but that is a general problem with this type process anyway (you can work around it by first deleting everything, breaking your site temporarily).
Composer helps you (among other things) "lock" all of the required libraries at some specific version. This is very useful when used with version control, tagged releases, and multiple environments (such as your development environment, a QA/Testing environment, and a Production environment). This helps ensure that each environment uses the exact same versions of the required vendor libraries.
If you were to try to use composer directly on your hosted server (i.e composer update), you would have to make sure that the php cli is available and that all other prerequisites for composer (including composer itself) were available. This is usually hard to count on for generic hosting providers so, you really are better off not trying to execute composer there.
As your project matures and uptime is important, you will want your development and deployment processes to mature as well. That's when you will realize many of the benefits of Composer.