I need a “Choose from existing content” popin

The plugin Custom Content Type Manager can do this.

A Custom Field of the Relation type is needed. When creating it, enable Is Repeatable, select the desired Output Filter and the Association.
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Make sure to enable Standardize Custom Fields for Pages in the Content Types tab of the plugin.

With all set, a Meta Box with the relationship/repeatable/sortable Custom Field will appear in all pages (looks like this will need a custom hack to only display it in $post->ID == homepage_ID).
This CF opens a thickbox where all post types can be searched/selected as a Relation (maybe another hack).


The plugin Advanced Custom Fields can do it all without hacks, but the Repeater is a premium add-on.