Hi DCNehs,
Thanks for your query in this community.
Unfortunately, it's not feasible to achieve what you requested, the
New function in the Document Library webpart is set as default, user could not add or edit the items inside of it.
So, in the scenario you described, the normal user have to open the document library to add the links item.
Given this kind of situation, if you have a strong concern on this, we would recommend you share your idea in
SharePoint UserVoice, which is a specific platform for us to collect user's request. Your suggestion will help us improve our Products and give users a better experience.
Or, if it’s convenient for you, you could use the Link webpart to meet your requirements in adding links directly at the home page.

Your understanding is highly appreciated.
Regards,
Alex Chen