My “Home” page is not showing under “pages”

In general “Home” refers to the main archive of blog (posts listing) page. This is a page that WP automatically creates and it is not an actual Page. The content of this page and how it has all the posts display is totally depend on the theme’s template.

Sometimes site owner want to set a Page as website’s front page, that page can be name whatever but usually it has “Home” as title.
https://codex.www.ads-software.com/Creating_a_Static_Front_Page

So what should I do? Should I make another page and transfer all my content to the new page, and delete the homepage? Or will the homepage always remain?

So what should I do?

It depends on what you are trying to achieve.

If you want a regular Page to be set as site’s front page, just make a Page and fill in content (that you normally do with Page), give it whatever title but for simplicity just title this Page as Home or Front, and then set this page as site front page, instruction is in the above link.

If you want to make your site front page do thing that normal Page could’t do, we need to create a template (WP php template), and this is not what normal users can do.

Should I make another page and transfer all my content to the new page, and delete the homepage?

The content as seen in your front home page right now is automatically pulled from posts and displayed in the fashion that is controlled by your theme’s template, so we can not move content in this page to a regular WP’s Page, also we can’t delete it.

Or will the homepage always remain?

Yes, but if you want to change how things displayed, you can make your own template or use a regular WP’s Page and set this page as front home page.