What can I use instead of gtk_container_set_focus_chain()?

The link you provided indicates:

Deprecated since: 3.24

For overriding focus behavior, use the GtkWidgetClass::focus signal.

which, of course, is almost fully undocumented (sigh). I have checked the demos and saw nothing related to this, sadly.

From this migration from GTK3 to GTK4 guide:

In GTK 4, any widget can have children (and GtkContainer is gone). There is new API to navigate the widget tree, for use in widget implementations: gtk_widget_get_first_child(), gtk_widget_get_last_child(), gtk_widget_get_next_sibling(), gtk_widget_get_prev_sibling().

This could be a good place to start.