Is is possible to add a live QuickSight visual onto a SharePoint page? If so, how?
You cannot add specific visual, but can add dashboard/s through embedding.
Ravi was correct at time of posting, but as of Aug 2022 you can embed individual visuals into other portals. From a dashboard (not an analysis) choose this:
Copy the embed code which is full HTML, but for SharePoint specifically you can use the Page Viewer Web Part, and only paste the URL from inside the iFrame in the code you copied. So if your embed code looks like this, only put the URL after the ‘src’ into the SP page viewer web part.
<iframe width="600" height="400" src="https://us-east-1.quicksight.aws.amazon.com/sn/embed/share/accounts/293424211206/dashboards/f628a8e4-e5f8-4196-99fd-470d70b5c7dc/sheets/f628a8e4-e5f8-4196-99fd-470d70b5c7dc_499f4505-f672-4286-94e3-fea1299f9f91/visuals/f628a8e4-e5f8-4196-99fd-470d70b5c7dc_65ece9e2-7b01-4391-9fed-121274867de6?directory_alias=quicksight-sales-demo"></iframe>