How do I add an application icon to the menu bar?

Nope, you add applications to your Dock for easy access. Review this support article on the many ways you can use the dock and how to add and remove the apps that are there.

Use the Dock on Mac - Apple Support

When using the Dock on the bottom of the screen, the only difference from what you are trying to accomplish is that you access it at the bottom instead of the top of the screen.

Ok. I did not recommend Bartender because there was some controversy about the app since the developer sold it to another company. I don't know enough about it to make a recommendation.

https://9to5mac.com/2024/06/04/bartender-acquired-unknown-developer/

As for the Dock, the only fixed Applications are the Finder and Trash. Running apps will appear there like you mentioned. By going to  > System Settings > Desktop & Dock, you can change the size of it to make it just about as small as the menu bar. You can also switch off the Suggested and Recent apps that may show there. I hide the Dock as you do, so it appears only when I move the cursor to the bottom of the screen.

Not trying to convince you of doing anything different than what you want, just making sure you know the options available to you.