Hi @kwilson good catch. When the total amount of links exceeds 5,000 the list functions normally, as it can hold up to 30 million items (SharePoint limits - Service Descriptions | Microsoft Learn).
However, the view threshold is indeed 5,000 items.
We added 2 more steps to the manual so that view threshold is not affecting the web part.
Please check those out, they can be applied now:
Under quick links manual: Quick Links Plus web part (with personalized links), expand section titled “Personalized links: Allow users to add link only visible to them”, and run additional steps: 12-14