I want to have a list of links in my SharePoint site that the user can edit and add to. Basically want it to look like this:
However when I add quick links to a drop zone in my page, I have to enter the links by hand. I must be missing something, this should be very easy to do.
From the screenshot looks like you are using modern pages and you might be out of luck here.
On modern pages, there is a quick links webpart which lets you add quick links but as you mentioned you have to edit the page and webpart properties to add new item (this is different from quick links app/list). There is a list webpart available on modern page (you need to see if it is rolled out to your tenant), which lets you add a custom list to a page. Unfortunately the current implementation does not provide a add new button.
Option 1: I would suggest you create a classic webpart page (Site Pages -> New -> Webpart page) which lets you add any list webpart (links list in your case) with complete experience you are looking for.
Option 2:If you have to use modern page, then create a custom list and add a hyperlink column to it (or design it as per your need). Then add a list webpart (modern version) and set the properties to show your list. You will not see add new button but you will see 'see all' button which will take you to the list directly where you can add new item.
