In case anyone runs into this in the future, here is how I solved it (with the help of a kind soul in the Rust Community Discord server).
It turns out that TextEditor captures the mouse event and MouseArea says “alright,  that’s handled then” and ignores it. So I wrapped the text_editor in a custom Widget (based on the code of MouseArea) that never reports an event as captured.
// main.rs
mouse_area(
    no_capture(
        text_editor(&state.srcbuf)
            .wrapping(iced::widget::text::Wrapping::None)
            .on_action(Message::ClickSrc)
            .width(500)
    )
)
.on_release(Message::ReleaseSrc)
// no_capture.rs
//! A container that ignores event captures.
//! Copied (and trimmed) from iced::widget::MouseArea.
use iced::advanced::layout;
use iced::advanced::mouse;
use iced::advanced::overlay;
use iced::advanced::renderer;
use iced::advanced::widget::{tree, Operation, Tree};
use iced::advanced::{Clipboard, Layout, Shell, Widget};
use iced::event::{self, Event};
use iced::{Element, Length, Rectangle, Size, Vector};
pub fn no_capture<'a, Message, Theme, Renderer>(
    content: impl Into<Element<'a, Message, Theme, Renderer>>,
) -> NoCapture<'a, Message, Theme, Renderer> {
    NoCapture {
        content: content.into(),
    }
}
/// Emit messages on mouse events.
#[allow(missing_debug_implementations)]
pub struct NoCapture<'a, Message, Theme = iced::Theme, Renderer = iced::Renderer> {
    content: Element<'a, Message, Theme, Renderer>,
}
impl<'a, Message, Theme, Renderer> NoCapture<'a, Message, Theme, Renderer> {
    /// Creates a [`NoCapture`] with the given content.
    pub fn new(content: impl Into<Element<'a, Message, Theme, Renderer>>) -> Self {
        NoCapture {
            content: content.into(),
        }
    }
}
#[derive(Default)]
enum State {
    #[default]
    None,
}
impl<Message, Theme, Renderer> Widget<Message, Theme, Renderer>
    for NoCapture<'_, Message, Theme, Renderer>
where
    Renderer: renderer::Renderer,
    Message: Clone,
{
    fn tag(&self) -> tree::Tag {
        tree::Tag::of::<State>()
    }
    fn state(&self) -> tree::State {
        tree::State::new(State::default())
    }
    fn children(&self) -> Vec<Tree> {
        vec![Tree::new(&self.content)]
    }
    fn diff(&self, tree: &mut Tree) {
        tree.diff_children(std::slice::from_ref(&self.content));
    }
    fn size(&self) -> Size<Length> {
        self.content.as_widget().size()
    }
    fn layout(
        &self,
        tree: &mut Tree,
        renderer: &Renderer,
        limits: &layout::Limits,
    ) -> layout::Node {
        self.content
            .as_widget()
            .layout(&mut tree.children[0], renderer, limits)
    }
    fn operate(
        &self,
        tree: &mut Tree,
        layout: Layout<'_>,
        renderer: &Renderer,
        operation: &mut dyn Operation,
    ) {
        self.content
            .as_widget()
            .operate(&mut tree.children[0], layout, renderer, operation);
    }
    fn on_event(
        &mut self,
        tree: &mut Tree,
        event: Event,
        layout: Layout<'_>,
        cursor: mouse::Cursor,
        renderer: &Renderer,
        clipboard: &mut dyn Clipboard,
        shell: &mut Shell<'_, Message>,
        viewport: &Rectangle,
    ) -> event::Status {
        self.content.as_widget_mut().on_event(
            &mut tree.children[0],
            event.clone(),
            layout,
            cursor,
            renderer,
            clipboard,
            shell,
            viewport,
        );
        event::Status::Ignored
    }
    fn mouse_interaction(
        &self,
        tree: &Tree,
        layout: Layout<'_>,
        cursor: mouse::Cursor,
        viewport: &Rectangle,
        renderer: &Renderer,
    ) -> mouse::Interaction {
        self.content.as_widget().mouse_interaction(
            &tree.children[0],
            layout,
            cursor,
            viewport,
            renderer,
        )
    }
    fn draw(
        &self,
        tree: &Tree,
        renderer: &mut Renderer,
        theme: &Theme,
        renderer_style: &renderer::Style,
        layout: Layout<'_>,
        cursor: mouse::Cursor,
        viewport: &Rectangle,
    ) {
        self.content.as_widget().draw(
            &tree.children[0],
            renderer,
            theme,
            renderer_style,
            layout,
            cursor,
            viewport,
        );
    }
    fn overlay<'b>(
        &'b mut self,
        tree: &'b mut Tree,
        layout: Layout<'_>,
        renderer: &Renderer,
        translation: Vector,
    ) -> Option<overlay::Element<'b, Message, Theme, Renderer>> {
        self.content
            .as_widget_mut()
            .overlay(&mut tree.children[0], layout, renderer, translation)
    }
}
impl<'a, Message, Theme, Renderer> From<NoCapture<'a, Message, Theme, Renderer>>
    for Element<'a, Message, Theme, Renderer>
where
    Message: 'a + Clone,
    Theme: 'a,
    Renderer: 'a + renderer::Renderer,
{
    fn from(
        area: NoCapture<'a, Message, Theme, Renderer>,
    ) -> Element<'a, Message, Theme, Renderer> {
        Element::new(area)
    }
}